True self-repelling motion above a general barrier

TR UE SELF-REPELLING MOTION ABO VE A GENERAL BARRIER LA URE MARÊCHÉ Abstra ct. The true self-rep elling motion is a con tinuous-time random process whic h w as introduced by Tóth and W erner in 1998 [32] to b e a limit for the “true” self-a voiding random w alk deﬁned b y Tóth in 1995 [28]. The construction of the true self-repelling motion in volv es an uncoun table system of coalescing Brownian motions starting from all points of the upp er half-plane, related to the Brownian web, but reﬂected and absorbed on a “barrier” whic h is the abscissa axis. In this w ork, w e consider muc h more general barriers, construct an uncountable system of coalescing Bro wnian motions reﬂected and absorb ed on these barriers, and the true self-rep elling motion associated to it. The extension of the pro ofs of Tóth and W erner to this more general case is surprisingly diﬃcult, esp ecially when the barrier is irregular. MSC2020: Primary 60K50; Secondary 60G99. Keyw ords: T rue self-rep elling motion, Brownian w eb, reﬂected and absorb ed Brownian motion. 1. Intr oduction The true self-r ep el ling motion is a con tin uous-time random pro cess which was introduced b y Tóth and W erner [32] as a candidate scaling limit for the discrete process kno wn as “true” self-avoiding walk with b ond r epulsion on Z . This “true” self-a voiding walk, whic h w as deﬁned b y Tóth in [28], describ es the p osition of a w alk er moving on Z in discrete time as follo ws. F or each edge of Z , w e call lo c al time on this edge the n umber of previous jumps of the w alk along the edge, and at eac h time, the w alk er can mov e along an edge of Z with a probabilit y proportional to exp( − β × ( lo cal time on the edge )) , β > 0 . In [28], Tóth asked the question of the scaling limit of the “true” self-a v oiding walk, but could not give a complete answer. Since then, this w alk and its generalizations hav e receiv ed a lot of atten tion [27, 29, 30, 31, 9, 10, 21, 18, 15, 19, 17, 3, 4], and the conv ergence of the w alk to the true self-rep elling motion was ﬁnally prov en b y K osygina and Peterson in [16]. The construction of the true self-rep elling motion is v ery complex and takes a large part of [32]. A sketc h is as follo ws. If ( x, h ) ∈ R × (0 , + ∞ ) , a r eﬂe cte d/absorb e d Br ownian motion (RAB) starting fr om ( x, h ) is a random function [ x, + ∞ ) → [0 , + ∞ ) whic h “at the left of 0” (on the in terv al [ x, 0] if x < 0 ) is a Brownian motion reﬂected on the abscissa axis, “at the right of 0” (on [0 , + ∞ ) if x ≤ 0 , on [ x, + ∞ ) if x > 0 ) is a Brownian motion absorb ed b y the abscissa axis, and whose v alue at x is h . Inspired b y the inﬁnite systems of coalescing Bro wnian motions in tro duced b y Arratia in [1, 2], Tóth and W erner [32] constructed a system of RABs starting from every p oin t of R × (0 , + ∞ ) so that any tw o of them coalesce when they meet, called system of forwar d lines . Since there are an uncountable n um b er of RABs, their construction is highly nontrivial. Afterwards, they built a system of b ackwar d lines , which are “forward lines going backw ards” (denoting Λ ∗ ( x,h ) ( . ) the backw ard lines, the Λ ∗ ( − x,h ) ( − . ) hav e the distribution of forw ard lines), but en t wined with the forward lines so a backw ard line can never cross a forward line. Again, the arguments are in tricate. They ﬁnally constructed the true self-rep elling motion ( X t ) t ≥ 0 b y pro ving that roughly , for an y t > 0 , there is a single ( x, h ) ∈ R × (0 , + ∞ ) so the integral of the function formed b y the forward and backw ard lines starting from ( x, h ) has v alue t , and they set X t = x . A num ber of prop erties of the true self-repelling motion were studied by Tóth and W erner in [32], and more w ere pro v en by Dumaz and Tóth in [6, 8, 7]. 1 2 LAURE MARÊCHÉ The system of forward lines built in [32] inspired the construction of the celebrated Brownian w eb by F on tes, Isopi, Newman and Ra vishank ar [12, 13]. The Brownian w eb is a system of Brownian motions starting from all p oin ts of R 2 whic h coalesce where they meet, seen in an appropriate top ology (see [25] for a review, see also the later works [11, 5]). It turned out to be a limiting ob ject for a lot of systems of coalescing random w alks (man y examples can b e found in Section 7.2 of [25]). How ev er, the construction of the true self-rep elling motion do es not require Brownian w eb to ols, hence w e will not use them here. A ﬁrst p ossible generalization of the true self-rep elling motion w as already mentioned in the original paper of Tóth and W erner [32]: instead of deﬁning the forward and bac kw ard lines as Brownian motions reﬂected/absorbed on the abscissa axis, they can b e reﬂected and absorb ed on another kind of “barrier”, a Bro wnian motion. A precise deﬁnition w as giv en in Section 10 of [32], but the changes needed in the proofs to accomo date for this diﬀeren t barrier were not written down. This particular generalization was studied further b y Newman and Ravishank ar in [23]. Ho w ever, one ma y need a true self-rep elling motion ab ov e a more general kind of “barrier” than the abscissa axis or the Brownian motion. F or example, if one studies the “true” self-a v oiding w alk of [28], but setting nonzero initial lo cal times, the resulting pro cess w ould b e exp ected to conv erge to a true self-rep elling motion abov e a “barrier” giv en b y the initial lo cal times proﬁle. The need for a true self-repelling motion abov e a diﬀerent barrier also arose in the study of the ne ar est-neighb or lifte d T ASEP , an interacting particle system in a class of mo dels designed to accelerate computations in Monte Carlo metho ds. In [20], Massoulié, Erignoux, T oninelli and Krauth pro v ed that the p ointer that gov erns the dynamics of this mo del can mov e like a “zero-temp erature” version of the “true” self-av oiding walk of [28], in whic h the w alk er alw a ys crosses the edge with the smallest lo cal time, with initial lo cal times dep ending on the initial particle conﬁguration. This w alk is also expected to con v erge to a true self-rep elling motion abov e a “barrier” giv en by the initial lo cal times proﬁle. In this pap er, w e construct a true self-rep elling motion ab o v e extremely general barriers. The extension of the construction of [32] to an y general barrier may seem ob vious at ﬁrst glance, but is m uc h more complex than it seems. Indeed, a general barrier, especially if it is irregular, ma y create a n um b er of indesirable eﬀects near it that w e ha v e to deal with. This fact was already noted b y Soucaliuc, Tóth and W erner [26], which considered deterministic barriers and constructed ﬁnite systems of forw ard and backw ard lines reﬂected/absorb ed on them (but neither inﬁnite systems nor true self-repelling motion). They had to require these barriers to be Lipschitz, and noted that their construction failed if they were not. In this w ork, we consider barriers whic h are random con tin uous functions λ : ( −∞ , + ∞ ) → R . W e also c ho ose a random χ ; our forw ard lines will b e Bro wnian motions reﬂected on λ at the left of χ and absorb ed b y λ at the righ t of χ (in the classical pro cess χ is 0 ); such pro cesses are called ( λ, χ ) -RABs. Without any other assumption on the barrier, w e are able to construct an uncoun table system of coalescing forward lines (Deﬁnition 5). W e then construct an uncountable system of backw ard lines ent wined with the forw ard lines (Deﬁnition 9) under an additional condition m uc h more lenient than the Lipschitz prop erty of [26]: we only ask the barrier λ to “b ehav e well immediately at the left of χ ” (see Deﬁnition 8). As w e explain in Remark 12, such a condition is necessary . Finally , we construct the true self-rep elling motion ab ov e the barrier λ stemming from these forward and backw ard lines, whic h w e call ( λ, χ ) -true self-r ep el ling motion (Deﬁnition 16), assuming a Brownian motion starting ab o ve λ alwa ys meets λ (see Deﬁnition 14), whic h is also a necessary condition. Our very general conditions on λ include the abscissa axis, Bro wnian motion, and deterministic Lipsc hitz barriers; in particular this work is also a written do wn construction of the true self-rep elling motion with a Bro wnian barrier. In addition to these constructions, w e pro v e a n um ber of properties for the forward lines, backw ard lines, and ( λ, χ ) -true self-rep elling motion, choosing the most relev ant ones while trying to contain the length of this w ork. W e also sho w some results on ( λ, χ ) -RABs whic h may be of independent in terest. TRUE SELF-REPELLING MOTION ABOVE A GENERAL BARRIER 3 This pap er unfolds as follo ws. In Section 2, w e state the deﬁnitions and properties of the systems of forw ard lines, bac kw ard lines, and the ( λ, χ ) -true self-rep elling motion. In Section 3, we gather some known results on Brownian motions, and original ones on ( λ, χ ) -RABs, which we use throughout the sequel. In Section 4 we prov e the prop erties of the forw ard lines. In Section 5 we show the prop erties of the bac kw ard lines. Finally , the construction and prop erties of the ( λ, χ ) -true self-repelling motion are pro v en in Section 6. 2. Definitions and pr oper ties In this section, we give the deﬁnitions and properties of all the ob jects constructed in this work: forw ard lines in Section 2.1, backw ard lines in Section 2.2, and ( λ, χ ) -true self-rep elling motion in Section 2.3. 2.1. F orward lines. This section is dev oted to the deﬁnition and properties of the forward lines. In order to construct these lines, we need sev eral auxiliary deﬁnitions. W e begin b y deﬁning our general barriers in Deﬁnition 1. W e then in tro duce the ( λ, χ ) -RABs as Brownian motions reﬂected by λ at the left of χ and absorb ed b y λ at the righ t of χ , in Deﬁnition 2. Then w e deﬁne ﬁnite families of indep endent coalescing ( λ, χ ) -RABs, or ( λ, χ ) -FICRABs, in Deﬁnition 4 and use this deﬁnition to construct the forward lines, which are an inﬁnite family of indep endent coalescing ( λ, χ ) -RABs (Deﬁnition 5). W e then state properties of the forw ard lines in Theorems 6 and 7. Deﬁnition 1. A b arrier is a couple ( λ, χ ) where λ : R 7→ R is a random contin uous function and χ is a real random v ariable. It ma y seem more natural to consider λ alone as the barrier, but later the prop erties w e ask of a “nice barrier” will dep end on χ , so we chose to include the latter in the deﬁnition of the barrier. In what follows, we will assume ( λ, χ ) is a barrier, and w e write P λ,χ for the probability conditionally to ( λ, χ ) . W e denote R 2 λ = { ( x, h ) ∈ R 2 | h > λ ( x ) } . Bro wnian motions will alwa ys ha ve the same v ariance v > 0 . Deﬁnition 2. F or an y ( x, h ) ∈ R 2 , the Br ownian motion r eﬂe cte d/absorb e d on ( λ, χ ) , or ( λ, χ ) -RAB , starting fr om ( x, h ) is a process ( R y ) y ≥ x deﬁned as follows. Let x ′ ≤ x , let ( W y ) y ≥ x ′ a Bro wnian motion indep endent from ( λ, χ ) , w e will say ( R y ) y ≥ x is driven by ( W y ) y ≥ x ′ . If h ≤ λ ( x ) , then ( R y ) y ≥ x is not deﬁned. If h > λ ( x ) : • If x ≥ χ , then ( R y ) y ≥ x is just a Brownian motion starting from h absorb ed by λ . If we denote Y = inf { y ≥ x | W y − W x + h = λ ( y ) } , then for y ∈ [ x, Y ] we set R y = W y − W x + h and for y ≥ Y we ha v e R y = λ ( y ) . • If x < χ , on [ x, χ ) , ( R y ) y ≥ x is a Bro wnian motion reﬂected on λ starting from h , and on [ χ, + ∞ ) , ( R y ) y ≥ x is a Brownian motion absorbed b y λ . If w e denote Y = inf { y ∈ [ x, χ ] | W y − W x + h = λ ( y ) } , for y ∈ [ x, χ ] , if y ≤ Y w e set R y = W y − W x + h and if y ≥ Y w e set R y = W y + sup Y ≤ z ≤ y ( λ ( z ) − W z ) . If we denote Y ′ = inf { y ≥ χ | W y − W χ + R χ = λ ( y ) } , then for y ∈ [ χ, Y ′ ] w e ha v e R y = W y − W χ + R χ and for y ≥ Y ′ w e hav e R y = λ ( y ) . R emark 3 . In most of the paper, it will b e con v enient that the ( λ, χ ) -RABs starting from ( x, λ ( x )) , x ∈ R stay undeﬁned. Ho w ever, at some p oints we will still need the notion of ( λ, χ ) -RABs starting from suc h a p oint. W e will call them b arrier-starting ( λ, χ ) -RAB starting fr om ( x, λ ( x )) , and deﬁne them as follows. Let x ′ ≤ x , let ( W y ) y ≥ x ′ a Bro wnian motion independent from ( λ, χ ) . If x ≥ χ , R y = λ ( y ) for any y ≥ x . If x < χ , for y ∈ [ x, χ ] , R y = W y + sup x ≤ z ≤ y ( λ ( z ) − W z ) and for y ≥ χ , R y is deﬁned as for standard ( λ, χ ) -RABs. W e no w construct ﬁnite families of independent coalescing ( λ, χ ) -RABs. W e will do this inductiv ely: if w e already ha v e a family of n − 1 pro cesses, w e build the n -th b y considering a ( λ, χ ) -RAB indep endent from the previous ones and following it un til it meeets one of them, after whic h w e follow the process it met. 4 LAURE MARÊCHÉ Deﬁnition 4. F or an y p ∈ N ∗ , an y ( x 1 , h 1 ) , ..., ( x p , h p ) ∈ R 2 , a ﬁnite family of indep endent c o alescing ( λ, χ ) -RABs , or ( λ, χ ) -FICRAB , starting fr om ( x 1 , h 1 ) , ..., ( x p , h p ) is a family (( C 1 ,y ) y ≥ x 1 , ..., ( C p,y ) y ≥ x p ) deﬁned as follows. Let ( R 1 ,y ) y ≥ x 1 ,..., ( R p,y ) y ≥ x p b e ( λ, χ ) -RABs driv en b y independent Brownian motions and starting resp ectively from ( x 1 , h 1 ) , ..., ( x p , h p ) . F or i ∈ { 1 , ..., p } , if ( R i,y ) y ≥ x i is not deﬁned, ( C i,y ) y ≥ x i is not deﬁned. W e set ( C 1 ,y ) y ≥ x 1 = ( R 1 ,y ) y ≥ x 1 , and for j ∈ { 2 , ..., p } we deﬁne by induction ω j = inf { x ≥ x j | R j,x ∈ { C 1 ,x , ..., C j − 1 ,x }} , ν j = min { k ∈ { 1 , ..., j − 1 } | R j,ω j = C k,ω j } , and C j,x = R j,x for x ∈ [ x j , ω j ] , C j,x = C ν j ,x for x ∈ [ ω j , + ∞ ) . The distribution of ( λ, χ, ( C 1 ,y ) y ≥ x 1 , ..., ( C p,y ) y ≥ x p ) actually do es not depend on the order in whic h the coalescence is p erformed. W e can no w deﬁne the system of forw ard lines. Let D b e the set of dy adic rational num bers. Deﬁnition 5. A system of forwar d lines ab ove ( λ, χ ) is a family of random maps (Λ ( x,h ) ( . )) ( x,h ) ∈ R 2 deﬁned as follows. Let { ( ˜ x n , ˜ h n ) } n ∈ N b e an enumeration of D 2 . (Λ ( ˜ x n , ˜ h n ) ( . )) n ∈ N is constructed inductively as a coun table family of indep enden t coalescing ( λ, χ ) -RABs starting from ( ˜ x n , ˜ h n ) , n ∈ N . Then, for any ( x, h ) ∈ R 2 , if h ≤ λ ( x ) then Λ ( x,h ) ( . ) is not deﬁned, and if h > λ ( x ) , then Λ ( x,h ) ( . ) is deﬁned on [ x, + ∞ ) by Λ ( x,h ) ( y ) = sup { Λ ( ˜ x n , ˜ h n ) ( y ) | ˜ x n < x, Λ ( ˜ x n , ˜ h n ) ( x ) < h } (it can b e sho wn as in [32] that almost surely , for all ( x, h ) ∈ R 2 λ this set is not empt y , and this deﬁnition is compatible with the deﬁnition of the Λ ( ˜ x n , ˜ h n ) ). In what follows, (Λ ( x,h ) ( . )) ( x,h ) ∈ R 2 will b e a system of forward lines ab o ve ( λ, χ ) . The follo wing theorem states a ﬁrst set of properties of this system and justiﬁes they form “an inﬁnite system of indep endent coalescing ( λ, χ ) -RABs”. Theorem 6. The fol lowing pr op erties hold. • F or any p ∈ N ∗ , if ( x 1 , h 1 ) , ..., ( x p , h p ) ∈ R 2 , then (Λ ( x 1 ,h 1 ) , ..., Λ ( x p ,h p ) ) is a ( λ, χ ) -FICRAB starting fr om ( x 1 , h 1 ) , ..., ( x p , h p ) . • A lmost sur ely, for al l ( x, h ) ∈ R 2 λ , then we have Λ ( x,h ) ( x ) = h . • A lmost sur ely, for al l ( x, h ) , ( x ′ , h ′ ) ∈ R 2 λ , if max( x, x ′ ) ≤ y ≤ z , then Λ ( x,h ) ( y ) < Λ ( x ′ ,h ′ ) ( y ) implies Λ ( x,h ) ( z ) ≤ Λ ( x ′ ,h ′ ) ( z ) . • A lmost sur ely, for any x ≤ y , the mapping : h 7→ Λ ( x,h ) ( y ) is non-de cr e asing and left-c ontinuous on ( λ ( x ) , + ∞ ) . In addition, these pr op erties char acterize the law of (Λ ( x,h ) ) ( x,h ) ∈ R 2 . W e now state another set of imp ortan t properties of the forw ard lines. F or any x < y in R , w e denote M ( x, y ) = { Λ ( z ,h ) ( y ) | z < x, ( z , h ) ∈ R 2 λ } , the “trace” at y of the forw ard lines starting b efore x . W e also denote M ( x ) = M ( x, x ) . Theorem 7 (Prop erties of the forward lines) . The fol lowing holds almost sur ely. • F or al l x ∈ R , M ( x ) is dense in [ λ ( x ) , + ∞ ) . • F or any x < y , M ( x, y ) is lo c al ly ﬁnite and unb ounde d. • F or al l ( x, h ) ∈ R 2 λ , ε > 0 , ther e exists n ∈ N so that ˜ x n < x , Λ ( ˜ x n , ˜ h n ) ( x ) ∈ ( h − ε, h ) and for y ≥ x + ε , Λ ( x,h ) ( y ) = Λ ( ˜ x n , ˜ h n ) ( y ) . • F or al l x < y , M ( x, y ) = { Λ ( ˜ x n , ˜ h n ) ( y ) | n ∈ N , ˜ x n < x, ( ˜ x n , ˜ h n ) ∈ R 2 λ } . • F or any ( x, h ) ∈ R 2 λ the function : y 7→ Λ ( x,h ) ( y ) is c ontinuous on [ x, + ∞ ) . The pro of of Theorems 6 and 7 are treated in Section 4. 2.2. Bac kw ard lines. In this section we state the deﬁnition of the bac kw ard lines (Deﬁnition 9) and an additional c haracterization (Proposition 10). W e then give the main prop erty of the backw ard lines, Theorem 11, which claims TRUE SELF-REPELLING MOTION ABOVE A GENERAL BARRIER 5 they behav e lik e forw ard lines “oriented bac kw ards”. W e also state a result on the num b er of forw ard and backw ard lines “incoming at a giv en p oin t”, Prop osition 13. W e b egin by formulating the additional prop ert y on ( λ, χ ) which is needed to construct the bac kw ard lines. Deﬁnition 8. The barrier ( λ, χ ) is said to be nic e if almost surely , there does not exist ε > 0 so that for an y x ∈ [ χ − ε, χ ] w e ha v e λ ( x ) − λ ( χ ) ≤ − √ χ − x . In the rest of this section and in Section 2.3 w e will alw a ys assume ( λ, χ ) nice. W e can no w construct the bac kward lines. Deﬁnition 9. F or any barrier ( λ, χ ) and system of forward lines (Λ ( x,h ) ( . )) ( x,h ) ∈ R 2 ab o ve ( λ, χ ) , the system of b ackwar d lines ab ove ( λ, χ ) is the family of random maps (Λ ∗ ( x,h ) ( . )) ( x,h ) ∈ R 2 deﬁned as follows. F or any ( x, h ) ∈ R 2 , if h ≤ λ ( x ) then Λ ∗ ( x,h ) ( . ) is not deﬁned, and if h > λ ( x ) w e deﬁne Λ ∗ ( x,h ) : ( −∞ , x ] 7→ R b y Λ ∗ ( x,h ) ( y ) = sup { Λ ( ˜ x n , ˜ h n ) ( y ) | ˜ x n < y , Λ ( ˜ x n , ˜ h n ) ( x ) < h } , whic h is set to λ ( y ) if the set is empty . The following characterization of the backw ard lines is easy to see. Prop osition 10. Almost sur ely, for al l ( x, h ) ∈ R 2 λ , y < x , we have Λ ∗ ( x,h ) ( y ) = sup { h ′ > λ ( y ) | Λ ( y ,h ′ ) ( x ) < h } , and for al l ( x, h ) ∈ R 2 λ , y > x we have Λ ( x,h ) ( y ) = sup { h ′ > λ ( y ) | Λ ∗ ( y ,h ′ ) ( x ) < h } (wher e the sup is set to λ ( y ) when the set is empty). The most important prop ert y of the system of bac kward lines is that it is “a system of forward lines oriented bac kw ards”, which is the follo wing. Theorem 11. (Λ ∗ ( − x,h ) ( − . )) ( x,h ) ∈ R 2 is a system of forwar d lines ab ove ( λ ( − . ) , − χ ) . The pro of of Theorem 11 can b e found in Section 5. R emark 12 . Theorem 11 will not hold if w e only assume λ con tin uous. Indeed, if we c ho ose χ = 0 and λ ( t ) = −| t | 1 / 3 for t ≥ 0 , whic h do not satisfy Deﬁnition 8, w e will see the bac kw ard lines may hit (0 , λ (0)) but not b e absorb ed b y λ there as they should, hence they do not hav e the correct distribution. This can be seen as follows. Results in Section 4.1 of [26] imply a ( λ, 0) -RAB starting at the left of 0 can hit the point (0 , λ (0)) with p ositive probability , so w e ha ve a p ositive probabilit y to get Λ ( ˜ x n , ˜ h n ) (0) = λ (0) for some n ∈ N . If this happ ens, for any x > 0 , h > λ ( x ) , w e hav e Λ ( ˜ x n , ˜ h n ) ( x ) = λ ( x ) < h , hence it w ould b e p ossible to ha ve Λ ∗ ( x,h ) (0) = λ (0) , but for ˜ x n < y < 0 , Λ ∗ ( x,h ) ( y ) ≥ Λ ( ˜ x n , ˜ h n ) ( y ) whic h can b e strictly ab ov e λ ( y ) , hence Λ ∗ ( x,h ) w ould hit (0 , λ (0)) , but not be absorb ed there. In Deﬁnition 8, we did not try to giv e an optimal condition, instead settling for a condition easy to state and to w ork with. This condition is satisﬁed if λ is the abscissa axis, a Bro wnian motion indep endent of χ , or a Lipsc hitz function as in [26]. W e will also state a prop erty of the systems of forw ard lines and backw ard lines, concerning the p ossible “n um b er of forw ard and bac kw ard lines incoming at a point”. In order to do that, we need to introduce some notation. F or an y ( x, h ) ∈ R 2 λ , we deﬁne the “n umber of forward lines incoming at ( x, h ) ” as I ( x, h ) = lim y → x,y x card( { p ∈ N | ∃ ( x 1 , h 1 ) , ..., ( x p , h p ) ∈ R 2 λ suc h that ∀ i ∈ { 1 , ..., p } , x i ≥ y , Λ ∗ ( x i ,h i ) ( x ) = h, ∀ z ∈ ( x, y ] , Λ ∗ ( x 1 ,h 1 ) ( z ) < · · · < Λ ∗ ( x p ,h p ) ( z ) } ) . F or an y ( x, h ) ∈ R 2 λ , the pair of integers [ I ( x, h ) , I ∗ ( x, h )] is called the typ e of ( x, h ) . The follo wing result sets heavy restrictions on the p ossible types of the points. 6 LAURE MARÊCHÉ Prop osition 13. W e have the fol lowing r esults on typ es. • F or any ( x, h ) ∈ R 2 , almost sur ely if h > λ ( x ) then ( x, h ) is of typ e [0 , 0] . • F or any x ∈ R , almost sur ely for any h > λ ( x ) , the p oint ( x, h ) is of typ e [0 , 0] , [1 , 0] or [0 , 1] . • A lmost sur ely for al l ( x, h ) ∈ R 2 λ , the p oint ( x, h ) is of typ e [0 , 0] , [1 , 0] , [0 , 1] , [1 , 1] , [2 , 0] or [0 , 2] . Prop osition 13 can b e prov en in the same w ay as Proposition 2.4 of [32] giv en the results in Section 5.4. 2.3. ( λ, χ ) -true self-rep elling motion. In this section, we b egin b y deﬁning the ( λ, χ ) -true self-rep elling motion (Prop osition 15 and Deﬁnition 16) and stating a ﬁrst set of its prop erties in Proposition 17. W e then construct a “lo cal time” for the ( λ, χ ) -true self-rep elling motion (Deﬁnition 19 and Theorem 20) and study its prop erties, in particular its relationship with the lines system (Prop osition 21 and Theorem 23). All the pro ofs are postp oned to Section 6. W e b egin by stating the additional condition on the barrier necessary to construct the ( λ, χ ) -true self-repelling motion. Deﬁnition 14. A barrier ( λ, χ ) is called a go o d b arrier if it is nice and when for all ( x, h ) ∈ R 2 , if ( W y ) y ∈ R is a t w o-sided Bro wnian motion indep enden t from ( λ, χ ) with W x = h , then P ( h > λ ( x ) , ∀ y > x, W y > λ ( y )) = 0 and P ( h > λ ( x ) , ∀ y < x, W y > λ ( y )) = 0 . In the rest of this section, ( λ, χ ) will alwa ys b e a go o d barrier. W e now need to deﬁne sev eral notations. As in [32], the notation ¯ Λ will encompass b oth forw ard and backw ard lines, as follows. F or any ( x, h ) ∈ R 2 , if h ≤ λ ( x ) then ¯ Λ ( x,h ) ( . ) is not deﬁned, and if h > λ ( x ) then ¯ Λ ( x,h ) : R 7→ R is deﬁned th us: for all y ∈ R , we set ¯ Λ ( x,h ) ( y ) = Λ ( x,h ) ( y ) if y ≥ x and ¯ Λ ( x,h ) ( y ) = Λ ∗ ( x,h ) ( y ) if y < x . This allo ws us to introduce, for any ( x, h ) ∈ R 2 λ , the quantit y T ( x, h ) = R + ∞ −∞ ( ¯ Λ ( x,h ) ( y ) − λ ( y ))d y . The additional prop ert y of Deﬁnition 14 is used to enforce the fact that almost surely , for all ( x, h ) ∈ R 2 λ , T ( x, h ) is ﬁnite (see Lemma 51). F or any t ≥ 0 , we deﬁne the set P t = T ε> 0 { ( x, h ) ∈ R 2 λ , T ( x, h ) ∈ ( t − ε, t + ε ) } (where A denotes the closure of a set A ). W e then ha v e the follo wing. Prop osition 15. Almost sur ely, for any t ∈ [0 , + ∞ ) , P t is a singleton. Deﬁnition 16. F or any t ≥ 0 , w e denote P t = { ( X t , H t ) } . The pro cess ( X t ) t ≥ 0 is called ( λ, χ ) -true self-r ep el ling motion . W e no w state a ﬁrst set of properties for the process (( X t , H t )) t ≥ 0 . Prop osition 17. The fol lowing holds almost sur ely. • F or al l t ∈ [0 , + ∞ ) , we have H t ≥ λ ( X t ) . • ( X t , H t ) t ≥ 0 is c ontinuous. • F or any x ∈ R , the set { t ≥ 0 | X t = x } is unb ounde d. R emark 18 . The classical true self-repelling motion deﬁned in [32] has the following inv ariance prop erties: ( X t , H t ) t ≥ 0 has the same distribution as ( − X t , H t ) t ≥ 0 and as ( a 2 / 3 X t/a , a 1 / 3 H t/a ) t ≥ 0 . W e cannot exp ect such inv ariance prop erties in our more general con text since λ will not necessarily hav e them. W e no w construct the “lo cal time” ( L t ( . )) t ≥ 0 of the pro cess ( X t ) t ≥ 0 and state some of its prop erties. Deﬁnition 19. F or any t ≥ 0 , x ∈ R , we deﬁne L t ( x ) = sup { h > λ ( x ) | T ( x, h ) < t } , setting the sup to λ ( x ) if the set is empty . Theorem 20. ( L t − λ ) t ≥ 0 is the lo cal time of the pr o c ess ( X t ) t ≥ 0 in the sense that almost sur ely, for any Bor el set A ⊂ R , t ≥ 0 , we have R t 0 1 { X s ∈ A } d s = R A ( L t ( x ) − λ ( x ))d x . TRUE SELF-REPELLING MOTION ABOVE A GENERAL BARRIER 7 Prop osition 21. The fol lowing holds almost sur ely. • F or al l t ≥ 0 , x ∈ R , L t ( x ) < + ∞ . • F or any x ∈ R , the function t 7→ L t ( x ) is non-de cr e asing and c ontinuous on [0 , + ∞ ) . • F or al l t ≥ 0 , H t = L t ( X t ) . R emark 22 . In [32], for any t ≥ 0 , the function L t has compact supp ort. This cannot be the case here, since L 0 = λ whic h does not necessarily has compact supp ort. Ho wev er, Lemmas 51 and 52 allo w to prov e that almost surely for all t ≥ 0 , the function L t − λ has compact supp ort. W e no w give a “Ra y-Knigh t Theorem” which shows a relationship betw een the local times ( L t ( . )) t ≥ 0 and the lines system. F or all ( x, h ) ∈ R 2 with h ≥ λ ( x ) , w e deﬁne T + ( x, h ) = lim ε → 0 ,ε> 0 T ( x, h + ε ) (which exists since T ( x, . ) is increasing on ( λ ( x ) , + ∞ ) ). Then the following stems directly from the deﬁnitions. Theorem 23 (Ra y-Knigh t Theorem) . The fol lowing holds almost sur ely. • F or al l ( x, h ) ∈ R 2 λ , y ∈ R , we have L T ( x,h ) ( y ) = ¯ Λ ( x,h ) ( y ) . • F or al l ( x, h ) ∈ R 2 with h ≥ λ ( x ) , we have L T + ( x,h ) ( y ) = lim ε → 0 ,ε> 0 ¯ Λ ( x,h + ε ) ( y ) . 3. Br o wnian motion and ( λ, χ ) -RAB In this section w e gather a set of results on Brownian motion and ( λ, χ ) -RAB whic h are used thoughout this pap er. The Brownian motion results are not original and are only giv en here for reference, but the ( λ, χ ) -RAB results are new to the author’s kno wledge and ma y b e of indep enden t interest. 3.1. Bro wnian motion results. The ﬁrst tw o lemmas give b ounds on the ﬂuctuations of a Brownian motion, an upp er bound for Lemma 24 and a lo w er bound for Lemma 25. Lemma 26 states an upp er bound on the probability three Brownian motions do not in tersect. Lemma 24. L et x ∈ R , ( W y ) y ≥ x a Br ownian motion (with varianc e v ), x ≤ y 1 < y 2 and a > 0 . Then P ( ∃ y ∈ [ y 1 , y 2 ] , | W y − W y 1 | ≥ a ) ≤ 2 √ 2 v ( y 2 − y 1 ) a √ π exp( − a 2 2 v ( y 2 − y 1 ) ) . F or a proof, see for example Remark 2.22 of [22]. Lemma 25. L et x ∈ R , ( W y ) y ≥ x a Br ownian motion (with varianc e v ), x ≤ y 1 < y 2 and a > 0 . P (max y ∈ [ y 1 ,y 2 ] ( W y − W y 1 ) ≤ a ) ≤ 2 a √ 2 π v ( y 2 − y 1 ) . Lemma 25 comes from the fact that the maxim um of the Bro wnian motion on an interv al is the absolute v alue of a Gaussian random v ariable (see for example Theorem 2.21 of [22]). The next lemma gives an upper b ound on the probability three Bro wnian motions do not intersect. It is not original, but it is surprisingly diﬃcult to ﬁnd a pro of sp elled out in the litterature, hence w e give an argumen t. Lemma 26. Ther e exists a c onstant ˜ C > 0 (dep ending on v ) so that for any ( x, h ) ∈ R 2 , ε > 0 , δ > 0 , if ( W 1 ( y )) y ≥ x , ( W 2 ( y )) y ≥ x , ( W 3 ( y )) y ≥ x ar e indep endent Br ownian motions so that W 1 ( x ) = h , W 2 ( x ) = h + δ , W 3 ( x ) = h + 2 δ , if Y = inf { y ≥ x | ∃ i, j ∈ { 1 , 2 , 3 } , W i ( y ) = W j ( y ) } , then P ( Y ≥ x + ε ) ≤ ˜ C ( δ √ ε ) 3 . Pr o of. Deﬁning γ ( t ) = 2(1 − R t −∞ 1 √ 2 π e − s 2 / 2 ds ) for an y t ∈ R , Theorem 3.1 of [24] states that P ( Y ≤ x + ε ) = 2 γ ( δ √ 2 v ε ) − γ ( 2 δ √ 2 v ε ) (note that we cannot use directly their Corollary 3.2 (ii), since it only gives an equiv alent for δ → 0 , ε ﬁxed). 8 LAURE MARÊCHÉ No w, γ ′ ( t ) = − 2 √ 2 π e − t 2 / 2 , γ ′′ ( t ) = 2 t √ 2 π e − t 2 / 2 , γ ′′′ ( t ) = 2 √ 2 π e − t 2 / 2 − 2 t 2 √ 2 π e − t 2 / 2 , hence γ ( t ) = 1 − 2 √ 2 π t + 1 3 √ 2 π t 3 + o ( t 3 ) , so when δ √ ε is small enough, P ( Y ≤ x + ε ) = 2 − 4 √ 2 π δ √ 2 v ε + 2 3 √ 2 π ( δ √ 2 v ε ) 3 − 1 + 2 √ 2 π 2 δ √ 2 v ε − 1 3 √ 2 π ( 2 δ √ 2 v ε ) 3 + o (( δ √ ε ) 3 ) = 1 − 6 3 √ 2 π ( δ √ 2 v ε ) 3 + o (( δ √ ε ) 3 ) , so P ( Y > x + ε ) = 1 2 √ π v 3 ( δ √ ε ) 3 + o (( δ √ ε ) 3 ) , whic h suﬃces to ﬁnd ˜ C > 0 so that P ( Y > x + ε ) ≤ ˜ C ( δ √ ε ) 3 for all v alues of δ √ ε . Moreo v er, the probability t wo of the W i meet precisely at x + ε is 0, so P ( Y = x + ε ) = 0 , whic h ends the pro of of the lemma. □ 3.2. ( λ, χ ) -RAB results. In this section, we ﬁrst pro ve an upp er b ound on the ﬂuctuations of a ( λ, χ ) -RAB as long as it sta ys aw a y from the barrier (Lemma 27). W e then study the atoms of the marginals of ( λ, χ ) -RABs, showing the marginal at y has no atom except p erhaps at λ ( y ) (Lemma 28) and that if ( λ, χ ) is nice, the marginal at χ has no atom at λ ( χ ) (Lemma 29). Finally , we prov e that it is imp ossible for a ( λ, χ ) -RAB to stay equal to λ on an en tire in terv al at the left of χ (Lemma 30). Lemma 27. L et x ∈ R , ( R y ) y ≥ x a ( λ, χ ) -RAB, x ≤ y 1 < y 2 and a > 0 . Then P ( { R y 1 − a ≥ max y ∈ [ y 1 ,y 2 ] λ ( y ) } ∩ {∃ y ∈ [ y 1 , y 2 ] , | R y − R y 1 | ≥ a } ) ≤ 2 √ 2 v ( y 2 − y 1 ) a √ π exp( − a 2 2 v ( y 2 − y 1 ) ) . Pr o of. This pro of uses the fact a ( λ, χ ) -RAB aw a y from the barrier b eha v es lik e a Brownian motion, and the b ound on the ﬂuctuations of a Brownian motion given in Lemma 24. W e denote b y ( W y ) y ≥ x the Bro wnian motion driving ( R y ) y ≥ x . W e notice that if R y 1 − a ≥ max y ∈ [ y 1 ,y 2 ] λ ( y ) and for all y ∈ [ y 1 , y 2 ] , | W y − W y 1 | < a , then R do es not meet λ in [ y 1 , y 2 ] , hence the increments of R in this in terv al are those of W , hence for all y ∈ [ y 1 , y 2 ] , | R y − R y 1 | < a . This implies P ( { R y 1 − a ≥ max y ∈ [ y 1 ,y 2 ] λ ( y ) } ∩ {∃ y ∈ [ y 1 , y 2 ] , | R y − R y 1 | ≥ a } ) ≤ P ( { R y 1 − a ≥ max y ∈ [ y 1 ,y 2 ] λ ( y ) } ∩ {∃ y ∈ [ y 1 , y 2 ] , | W y − W y 1 | ≥ a } ) ≤ P ( ∃ y ∈ [ y 1 , y 2 ] , | W y − W y 1 | ≥ a ) ≤ 2 √ 2 v ( y 2 − y 1 ) a √ π exp( − a 2 2 v ( y 2 − y 1 ) ) by Lemma 24. □ The following lemma sho ws the marginal at y of a ( λ, χ ) -RAB has no atoms except perhaps at λ ( y ) . Lemma 28. L et ( x, h ) ∈ R 2 and ( R y ) y ≥ x a ( λ, χ ) -RAB starting fr om ( x, h ) , or let x ∈ R and ( R y ) y ≥ x a b arrier- starting ( λ, χ ) -RAB starting fr om ( x, λ ( x )) , then for any y > x , we have P ( a > λ ( y ) , R y = a ) = 0 . Pr o of. W e use the fact that when a ( λ, χ ) -RAB is strictly abov e λ , it behav es like a Brownian motion which has no atoms. W e denote ( W z ) z ≥ x the Bro wnian motion driving ( R z ) z ≥ x . Then if a > λ ( y ) and R y = a , there exists some z ∈ ( x, y ) rational so that for y ′ ∈ [ z , y ] , w e ha v e R y ′ > λ ( y ′ ) , hence R y − R z = W y − W z , hence W y − W z = a − R z . This yields P ( a > λ ( y ) , R y = a ) ≤ P z ∈ ( x,y ) ∩ Q P ( W y − W z = a − R z ) = 0 since for z ∈ ( x, y ) , w e hav e W y − W z Gaussian indep endent from R z . □ The following lemma states the marginal at χ of a ( λ, χ ) -RAB has no atom at λ ( χ ) . In order to do that, we crucially need ( λ, χ ) to be a nice barrier. Lemma 29. L et ( λ, χ ) b e a deterministic nic e b arrier. L et ( x, h ) ∈ R 2 λ with x < χ , and let ( R y ) y ≥ x b e a ( λ, χ ) -RAB starting fr om ( x, h ) . Then P ( R χ = λ ( χ )) = 0 . Pr o of. W e denote by ( W y ) y ≥ x the Brownian motion driving ( R y ) y ≥ x , and Y = inf { y ≥ x | W y − W x + h = λ ( y ) } . If R χ = λ ( χ ) , there are tw o p ossibilities: either Y = χ , which implies W χ − W x = λ ( χ ) − h , whic h has probability 0, or Y < χ . Thus it is enough to prov e P ( R χ = λ ( χ ) , Y < χ ) = 0 . F urthermore, if R χ = λ ( χ ) and Y < χ , then W χ + sup Y ≤ y ≤ χ ( λ ( y ) − W y ) = λ ( χ ) , th us sup Y ≤ y ≤ χ (( λ ( y ) − λ ( χ )) − ( W y − W χ )) = 0 , hence λ ( y ) − λ ( χ ) ≤ W y − W χ for all Y ≤ y ≤ χ . Moreov er ( λ, χ ) is nice, hence there exists a strictly increasing sequence x n tending to χ so that λ ( x n ) − λ ( χ ) > − √ χ − x n . Therefore, if R χ = λ ( χ ) and Y < χ , there exists some N ∈ N so that for n ≥ N we TRUE SELF-REPELLING MOTION ABOVE A GENERAL BARRIER 9 ha v e W x n − W χ > − √ χ − x n . Denoting E = {∀ N ∈ N , ∃ n ≥ N , W x n − W χ ≤ − √ χ − x n } , we deduce that it is enough to pro v e that P ( E ) = 1 . In addition, E is in the germ σ -algebra of the Bro wnian motion ( W χ − y − W χ ) 0 ≤ z ≤ χ − x , hence P ( E ) = 0 or 1 (see for example Theorem 2.7 of [22]). F urthermore, P ( E ) = lim N → + ∞ P ( ∃ n ≥ N , W x n − W χ ≤ − √ χ − x n ) ≥ lim N → + ∞ P ( W x N − W χ ≤ − √ χ − x N ) , whic h is the probability a centered Gaussian random v ariable of v ariance v is smaller than -1, which is positive. Hence P ( E ) > 0 , th us P ( E ) = 1 , whic h ends the pro of. □ The last lemma of this section prov es that in the part of the space where it is reﬂected, a ( λ, χ ) -RAB cannot “stic k” to λ , in the sense that it cannot sta y equal to λ on a whole open in terv al. Lemma 30. L et ( λ, χ ) b e a deterministic b arrier. L et ( x, h ) ∈ R 2 λ with x < χ , and let ( R y ) y ≥ x b e a ( λ, χ ) -RAB starting fr om ( x, h ) . F or any x ≤ a < b ≤ χ , P ( ∀ y ∈ [ a, b ] , R y = λ ( y )) = 0 . Pr o of. W e will ﬁrst pro ve it is enough to ﬁnd some y 0 ∈ [ a, b ) so that there exist ε ∈ (0 , b − y 0 ) , κ ∈ R so that for all y ∈ [ y 0 , y 0 + ε ] we ha v e λ ( y ) − λ ( y 0 ) ≤ κ ( y − y 0 ) . Indeed, let ( W y ) y ≥ x b e the Bro wnian motion driving ( R y ) y ≥ x . F or an y y ∈ [ y 0 , y 0 + ε ] , we ha ve R y − R y 0 ≥ W y − W y 0 , hence R y − λ ( y ) ≥ W y − W y 0 + λ ( y 0 ) − λ ( y ) ≥ W y − W y 0 − κ ( y − y 0 ) . Moreo v er, the la w of iterated logarithm states that almost surely lim sup y → y 0 ,y >y 0 W y − W y 0 √ 2( y − y 0 ) ln(ln(1 / ( y − y 0 ))) = 1 , hence almost surely there exists y ∈ [ y 0 , y 0 + ε ] so that W y − W y 0 − κ ( y − y 0 ) > 0 , th us R y > λ ( y ) , which is enough. Consequen tly , w e only hav e to ﬁnd a suitable y 0 . If there exists y ∈ [ a, b ) so that λ has a lo cal maximum in [ y , b ] at y , then we set y 0 = y . If there is no suc h maximum, then λ is strictly increasing on [ a, b ] . Indeed, if there were a ≤ y < y ′ ≤ b with λ ( y ) ≥ λ ( y ′ ) , then λ w ould ha v e a global maxim um on [ y , y ′ ] whic h would b e a lo cal maxim um as ab ov e. F urthermore, since λ is increasing on [ a, b ] , it is diﬀeren tiable almost ev erywhere on [ a, b ] by Leb esgue’s Theorem (see Theorem 7.2 in [14]), so w e can c hoose a p oin t in [ a, b ) at whic h λ is diﬀerentiable, and this p oint is a suitable y 0 . □ 4. F or w ard lines: proof of Theorems 6 and 7 Theorems 6 and 7, whic h gather properties of the forw ard lines, were prov en in the case of the classical true self- rep elling motion in [32] (Theorem 2.1 and Prop osition 1.2). A large part of the proofs of [32] can b e extended to our setting, but the greater generalit y of our barriers requires c hanges at several p oin ts. Here w e detail these modiﬁcations, whic h are in their Lemma 8.1, Equation (8.22), Lemma 8.2 and Equation (8.49). 4.1. Lemma 8.1 of [32] . The pro of of part (i) of Lemma 8.1 in [32] relies on an estimate on the probability t w o RABs with close starting p oints meet quic kly , itself relying on the fact their barrier is constan t. T o comp ensate for the lac k of suc h an estimate for ( λ, χ ) -RABs, w e pro v e they will meet before hitting the barrier, when still b ehaving as Bro wnian motions. This leads us to replace part (i) of Lemma 8.1 of [32] b y part (i) of the follo wing lemma, whic h is a bit weak er that the result of [32], but is suﬃcien t. Moreov er, the random nature of the barrier requires an additional argumen t for part (ii) of Lemma 8.1 of [32], which is part (ii) of the following lemma. Lemma 31. L et ( x, h ) ∈ R 2 . (i) Ther e exists a deterministic se quenc e ( n ( k )) k ≥ 1 so that for any k ≥ 1 , ˜ x n ( k ) < x , lim k → + ∞ ˜ x n ( k ) = x , lim k → + ∞ ˜ h n ( k ) = h , and for al l ε > 0 , if λ ( x ) < h , when k is lar ge enough P λ,χ ( ∃ y ≥ x + ε, Λ ( x,h ) ( y )  = Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) ( y )) ≤ 7 √ π v 2 − k/ 2 . (ii) A lmost sur ely, if λ ( x ) < h , for al l ε > 0 , ther e exists k 0 ≥ 1 so for al l k ≥ k 0 , we have Λ ( x,h ) ( y ) = Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) ( y ) for al l y ≥ x + ε . 10 LAURE MARÊCHÉ Pr o of. (i) As in [32], w e “squeeze Λ ( x,h ) b et ween tw o families of lines Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) and Λ ( ˜ x m ( k ) , ˜ h m ( k ) ) whic h ha v e high probabilit y of coalescing b efore x + ε , so after x + ε , Λ ( x,h ) is Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) ”. W e c ho ose t w o sequences (( ˜ x n ( k ) , ˜ h n ( k ) )) k ≥ 1 and (( ˜ x m ( k ) , ˜ h m ( k ) )) k ≥ 1 in R 2 so that when k is large enough, ˜ x n ( k ) = ˜ x m ( k ) ∈ ( x − 5 − k , x ) , h − 2 · 2 − k ≤ ˜ h n ( k ) ≤ h − 2 − k and h + 2 − k ≤ ˜ h m ( k ) ≤ h + 2 · 2 − k . In this proof we assume λ ( x ) < h . W e now build “bad even ts” suc h that {∃ y ≥ x + ε, Λ ( x,h ) ( y )  = Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) ( y ) } ensures the o ccurrence of a bad ev en t. Since λ is con tin uous and λ ( x ) < h , there exists 0 < δ < ε so that for all y ∈ [ x − δ, x + δ ] , we hav e λ ( y ) < h − δ . W e consider only k large enough to hav e 4 · 2 − k ≤ δ . W e deﬁne the ev ents A 1 ,k = {∃ y ∈ [ ˜ x n ( k ) , x + 5 − k ] , Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) ( y ) ∈ ( h − 3 · 2 − k , h ) } and A 2 ,k = {∃ y ∈ [ ˜ x m ( k ) , x + 5 − k ] , Λ ( ˜ x m ( k ) , ˜ h ( k ) ) ( y ) ∈ ( h, h + 3 · 2 − k ) } . By Lemma 27, when k is large enough, (1) P λ,χ ( A 1 ,k ) ≤ 2 √ 4 v 5 − k 2 − k √ π exp  − 2 − 2 k 4 v 5 − k  ≤ exp  − 5 k v 4 k +1  and P ( A 2 ,k ) ≤ exp  − 5 k v 4 k +1  . In [32], the authors also study the even t { Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) ( x + ε )  = Λ ( ˜ x m ( k ) , ˜ h m ( k ) ) ( x + ε ) } , and prov e it has small probability b y estimating the probabilit y that t wo RABs coalesce. Since we cannot do this here, w e replace their ev ent with A 3 ,k = { Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) ( x + 2 − k )  = Λ ( ˜ x m ( k ) , ˜ h m ( k ) ) ( x + 2 − k ) } . The idea will b e to prov e that in [ x, x + 2 − k ] , the pro cesses Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) and Λ ( ˜ x m ( k ) , ˜ h m ( k ) ) will not go too far from their positions at x , so will not meet λ hence will b eha ve like t w o indep endent Bro wnian motions, th us since their starting p oin ts are close they will meet with high probability . Since 2 − k ≤ δ ≤ ε , w e ha v e {∃ y ≥ x + ε, Λ ( x,h ) ( y )  = Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) ( y ) } ⊂ A 1 ,k ∪ A 2 ,k ∪ A 3 ,k . Consequen tly (recalling (1)), to prov e (i), it is enough to pro v e (2) P λ,χ ( A c 1 ,k ∩ A c 2 ,k ∩ A 3 ,k ) ≤ 6 + 1 / 2 √ π v 2 − k/ 2 when k is large enough. W e now study P λ,χ ( A c 1 ,k ∩ A c 2 ,k ∩ A 3 ,k ) . W e can assume the pro cesses Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) and Λ ( ˜ x m ( k ) , ˜ h m ( k ) ) form a ( λ, χ ) -FICRAB built from the ( λ, χ ) -RABs ( R n ( k ) ( y )) y ≥ ˜ x n ( k ) , ( R m ( k ) ( y )) y ≥ ˜ x m ( k ) , themselves driv en by the independent Bro wnian motions ( W n ( k ) ( y )) y ≥ ˜ x n ( k ) and ( W m ( k ) ( y )) y ≥ ˜ x m ( k ) . Let A 4 ,k = { ( ˜ x n ( k ) , ˜ h n ( k ) ) , ( ˜ x m ( k ) , ˜ h m ( k ) ) ∈ R 2 λ , ∀ y ∈ [ x, x + 2 − k ] , W n ( k ) ( y ) − W n ( k ) ( x ) + Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) ( x )  = W m ( k ) ( y ) − W m ( k ) ( x ) + Λ ( ˜ x m ( k ) , ˜ h m ( k ) ) ( x ) } and A 5 ,k = {∃ y ∈ [ x, x + 2 − k ] , | W n ( k ) ( y ) − W n ( k ) ( x ) | > δ / 4 or | W m ( k ) ( y ) − W m ( k ) ( x ) | > δ / 4 } . W e assume A c 1 ,k ∩ A c 2 ,k o ccurs and A 4 ,k ∪ A 5 ,k do es not o ccur. Since A c 1 ,k ∩ A c 2 ,k o ccurs, Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) and Λ ( ˜ x m ( k ) , ˜ h m ( k ) ) ha v e not coalesced yet at x . Since A c 1 ,k ∩ A c 2 ,k ∩ A c 5 ,k o ccurs, for all y ∈ [ x, x + 2 − k ] w e ha v e R n ( k ) ( y ) > λ ( y ) thus R n ( k ) ( y ) = W n ( k ) ( y ) − W n ( k ) ( x ) + R n ( k ) ( x ) = W n ( k ) ( y ) − W n ( k ) ( x ) + Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) ( x ) , and similarly R m ( k ) ( y ) = W m ( k ) ( y ) − W m ( k ) ( x ) + Λ ( ˜ x m ( k ) , ˜ h m ( k ) ) ( x ) . Since A 4 ,k do es not o ccur, Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) and Λ ( ˜ x m ( k ) , ˜ h m ( k ) ) ha v e coalesced b efore x + 2 − k , th us b efore x + ε , hence A 3 ,k do es not o ccur. W e deduce P λ,χ ( A c 1 ,k ∩ A c 2 ,k ∩ A 3 ,k ) ≤ P λ,χ (( A 4 ,k ∪ A 5 ,k ) ∩ A c 1 ,k ∩ A c 2 ,k ) , so it is enough to pro v e P λ,χ (( A 4 ,k ∪ A 5 ,k ) ∩ A c 1 ,k ∩ A c 2 ,k ) ≤ 6+1 / 2 √ π v 2 − k/ 2 when k is large enough. W e now study P λ,χ ( A 4 ,k ∩A c 1 ,k ∩A c 2 ,k ) and P λ,χ ( A 5 ,k ) . By the standard estimate on Brownian motion given in Lemma 24, we ha v e P λ,χ ( A 5 ,k ) ≤ 16 √ 2 v 2 − k δ √ π exp( − δ 2 32 v 2 − k ) ≤ exp( − δ 2 2 k 32 v ) ≤ 1 2 √ π v 2 − k/ 2 when k is large enough. T o deal with A 4 ,k , w e notice that if ( ˜ x n ( k ) , ˜ h n ( k ) ) , ( ˜ x m ( k ) , ˜ h m ( k ) ) ∈ R 2 λ , then P λ,χ ( A 4 ,k | Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) ( x ) , Λ ( ˜ x m ( k ) , ˜ h m ( k ) ) ( x )) is the probabilit y TRUE SELF-REPELLING MOTION ABOVE A GENERAL BARRIER 11 that the Bro wnian motion W n ( k ) ( y ) − W n ( k ) ( x ) − ( W m ( k ) ( y ) − W m ( k ) ( x )) do es not reac h Λ ( ˜ x m ( k ) , ˜ h m ( k ) ) ( x ) − Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) ( x ) in [ x, x + 2 − k ] , and if A c 1 ,k ∩ A c 2 ,k o ccurs we get | Λ ( ˜ x m ( k ) , ˜ h m ( k ) ) ( x ) − Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) ( x ) | ≤ 6 · 2 − k . Therefore Lemma 25 implies P λ,χ ( A 4 ,k ∩ A c 1 ,k ∩ A c 2 ,k ) ≤ E λ,χ ( P λ,χ ( A 4 ,k | Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) ( x ) , Λ ( ˜ x m ( k ) , ˜ h m ( k ) ) ( x )) 1 {| Λ ( ˜ x m ( k ) , ˜ h m ( k ) ) ( x ) − Λ ( ˜ x n ( k ) , ˜ h n ( k ) ) ( x ) |≤ 6 · 2 − k } ) ≤ P (max y ∈ [ x,x +2 − k ] ( W n ( k ) ( y ) − W n ( k ) ( x ) − ( W m ( k ) ( y ) − W m ( k ) ( x ))) ≤ 6 · 2 − k ) ≤ 6 · 2 − k √ π v 2 − k = 6 √ π v 2 − k/ 2 , whic h tends to 0 when k tends to + ∞ . Consequen tly , P (( A 4 ,k ∪ A 5 ,k ) ∩ A c 1 ,k ∩ A c 2 ,k ) ≤ 6+1 / 2 √ π v 2 − k/ 2 when k is large enough, which ends the pro of of (i). (ii) If λ ( x ) < h , then by (1) and (2), w e hav e P λ,χ ( T ℓ ∈ N S k ≥ ℓ ( A 1 ,k ∪ A 2 ,k ∪ A 3 ,k ))) = 0 . Therefore P ( { λ ( x ) < h } ∩ T ℓ ∈ N S k ≥ ℓ ( A 1 ,k ∪ A 2 ,k ∪ A 3 ,k ))) = 0 , which suﬃces. □ 4.2. Equation (8.22) of [32] . T o understand Equation (8.22) of [32], we need to giv e some notation. Let p ∈ N ∗ , ( x 1 , h 1 ) , ..., ( x p , h p ) ∈ R 2 , w e denote (( C 1 ( y )) y ≥ x 1 , ..., ( C p ( y )) y ≥ x p ) a ( λ, χ ) -FICRAB starting from ( x 1 , h 1 ) , ..., ( x p , h p ) . Let r ∈ N ∗ , set a map j : { 1 , ..., r } 7→ { 1 , ..., p } . F or i ∈ { 1 , ..., r } , set y i > x j ( i ) and λ ( y i ) < a i < b i . F or eac h j ∈ { 1 , ..., p } , k ∈ N , let ( ˜ x n j ( k ) , ˜ h n j ( k ) ) ∈ D 2 so that ˜ x n j ( k ) ∈ ( x j − 5 − k , x j ) and ˜ h n j ( k ) ∈ ( h j − 2 · 2 − k , h j − 2 k ) . The equiv alent of Equation (8.22) of [32] is pro ving the follo wing. Prop osition 32. If ( x 1 , h 1 ) , ..., ( x p , h p ) ∈ R 2 λ , P λ,χ ( ∀ i ∈ { 1 , ..., r } , Λ ( ˜ x n j ( i ) ( k ) , ˜ h n j ( i ) ( k ) ) ( y i ) ∈ ( a i , b i )) c onver ges to P λ,χ ( ∀ i ∈ { 1 , ..., r } , C j ( i ) ( y i ) ∈ ( a i , b i )) when k tends to + ∞ . In [32], the only justiﬁcation given to this is that ( ˜ x n j ( k ) , ˜ h n j ( k ) ) conv erges to ( x j , h j ) for j ∈ { 1 , ..., p } . Here, since the barriers are more complex, we need more arguments. Pr o of of Pr op osition 32. W e will construct pro cesses (Λ ′ j,k ) 1 ≤ j ≤ p and ( C ′ j ) 1 ≤ j ≤ p whic h ha ve the same distribution as the (Λ ( ˜ x n j ( k ) , ˜ h n j ( k ) ) ) 1 ≤ j ≤ p and ( C j ) 1 ≤ j ≤ p , and so that the ﬁnite-dimensional marginals of (Λ ′ j,k ) 1 ≤ j ≤ p con v erge in probabilit y to those of ( C ′ j ) 1 ≤ j ≤ p . Let ( W j ( x )) x ≥ x j − 1 , j ∈ { 1 , ..., p } b e indep enden t Bro wnian motions. F or an y j ∈ { 1 , ..., p } , let R ′ j the ( λ, χ ) -RAB starting from ( x j , h j ) and driven b y W j . ( C ′ 1 , ..., C ′ p ) is then deﬁned as the ( λ, χ ) -FICRAB constructed from ( R ′ 1 , ..., R ′ p ) . F or all k ∈ N , j ∈ { 1 , ..., p } , let Λ R j,k the ( λ, χ ) -RAB starting from ( ˜ x n j ( i ) ( k ) , ˜ h n j ( i ) ( k ) ) and driven b y W j . (Λ ′ 1 ,k , ..., Λ ′ p,k ) is then deﬁned as the ( λ, χ ) -FICRAB constructed from (Λ R 1 ,k , ..., Λ R p,k ) . In the whole pro of of Proposition 32, we w ork conditionally to ( λ, χ ) . W e begin b y proving the following lemma, whic h states the Λ R j,k are close to the R ′ j with high probability . Lemma 33. F or any j ∈ { 1 , ..., p } , y 0 > max 1 ≤ i ≤ r y i , ε > 0 , P λ,χ ( ∃ x j ≤ y ≤ y 0 , R ′ j ( y ) − Λ R j,k ( y ) ∈ [0 , ε ]) tends to 0 when k tends to + ∞ . Pr o of. It is enough to prov e the lemma for ε small, so let y 0 > max 1 ≤ i ≤ r y i , j ∈ { 1 , ..., p } and ε > 0 small. Let ε ′ > 0 , it is enough to pro v e that P λ,χ ( ∃ x j ≤ y ≤ y 0 , R ′ j ( y ) − Λ R j,k ( y ) ∈ [0 , ε ]) ≤ ε ′ when k is large enough. Let k ∈ N large enough to ha v e 3 · 2 − k ≤ ε and 2 √ 2 v 5 − k 2 − k √ π exp( − 2 − 2 k 2 v 5 − k ) ≤ ε ′ / 3 . W e are going to construct ev en ts with small probability so that if none of them o ccurs, for all x j ≤ y ≤ y 0 , R ′ j ( y ) − Λ R j,k ( y ) ∈ [0 , ε ] . W e denote A 1 ,k = {∃ x ∈ [ ˜ x n j ( k ) , x j ] , W j ( x ) − W j ( ˜ x n j ( k ) ) ∈ [ − 2 − k , 2 − k ] } . Then, b y the classical estimate on Brownian motion recalled in Lemma 24, w e hav e P λ,χ ( A 1 ,k ) ≤ 2 √ 2 v 5 − k 2 − k √ π exp( − 2 − 2 k 2 v 5 − k ) ≤ ε ′ / 3 . F urthermore, since h j > λ ( x j ) and λ is 12 LAURE MARÊCHÉ con tin uous, when k is large enough, for all x ∈ [ x j − 5 − k , x j ] , λ ( x ) < h j − 3 · 2 − k . In the following we only consider k large enough for this to happ en. Then if A 1 ,k do es not o ccur, (3) ∀ y ∈ [ ˜ x n j ( k ) , x j ] , Λ R j,k ( y )  = λ ( y ) and Λ R j,k ( x j ) ∈ [ h j − 3 · 2 − k , h j ] , hence λ ( x j ) < R ′ j ( x j ) − 3 · 2 − k ≤ Λ R j,k ( x j ) ≤ R ′ j ( x j ) . Moreo v er, λ and W j are uniformly contin uous on [ x j , y 0 + 1] , so w e can choose 0 < δ 2 < 1 so that for y , y ′ ∈ [ x j , y 0 + 1] , if | y − y ′ | ≤ δ 2 then | λ ( y ) − λ ( y ′ ) | ≤ ε/ 2 , and denoting A 2 = {∃ y , y ′ ∈ [ x j , y 0 + 1] so that | y − y ′ | ≤ δ 2 and | W j ( y ) − W j ( y ′ ) | > ε/ 2 } , then P λ,χ ( A 2 ) ≤ ε ′ / 3 . W e no w need to deﬁne even ts ensuring “ Λ R j,k will b e absorb ed by λ almost at the same place as R ′ j ”. In order to do that, we will use Lemma 8.2 of [19]. It states that if a < b , if f : [ a, b ] 7→ R is a con tin uous function, if ( W ( y )) y ∈ [ a,b ] is a Brownian motion so that W ( a ) > f ( a ) almost surely , if for an y δ ∈ R we denote σ ( δ ) = inf { t ∈ [ a, b ] | W ( y ) ≤ f ( y ) + δ } (the inf b eing + ∞ when the set is empty), then σ ( δ ) conv erges in probabilit y to 0 when δ tends to 0 (Lemma 8.2 of [19] is stated for a = 0 , b = 1 , but easily extends to general a and b ). F or all δ ∈ R , w e denote Y ( δ ) = inf { y ∈ [ x j , y 0 ] | W j ( y ) − W j ( x j ) + h j ≤ λ ( y ) + δ } (the inf b eing + ∞ when the set is empt y), then since h j > λ ( x j ) , b y Lemma 8.2 of [19], Y ( δ ) conv erges in P λ,χ -probabilit y to Y (0) when δ tends to 0. Therefore w e can choose δ 3 > 0 so that P λ,χ ( Y (0) − Y ( δ 3 ) > δ 2 ) ≤ ε ′ / 6 . Here we will need to distinguish betw een the cases x j ≥ χ and x j < χ . If x j ≥ χ , we denote A 3 = { Y (0) − Y ( δ 3 ) > δ 2 } . If x j < χ and y 0 > χ (deﬁning this ev ent is not necessary if y 0 ≤ χ ), for all δ ∈ R , w e denote Y ′ ( δ ) = inf { y ∈ [ χ, y 0 ] | W j ( y ) − W j ( χ ) + R ′ j ( χ ) ≤ λ ( y ) + δ } (the inf being + ∞ when the set is empt y). Then b y the Lemma 8.2 of [19], P λ,χ ( Y ′ (0) − Y ′ ( δ ) > δ 2 | R ′ j ( χ )) 1 { R ′ j ( χ ) >λ ( χ ) } con v erges almost surely to 0 when δ tends to 0, hence P λ,χ ( R ′ j ( χ ) > λ ( χ ) , Y ′ (0) − Y ′ ( δ ) > δ 2 ) tends to 0 when δ tends to 0. W e th us c hoose δ 4 > 0 so that P λ,χ ( R ′ j ( χ ) > λ ( χ ) , Y ′ (0) − Y ′ ( δ 4 ) > δ 2 ) ≤ ε ′ / 6 and set A 3 = { Y (0) − Y ( δ 3 ) > δ 2 } ∪ { R ′ j ( χ ) > λ ( χ ) , Y ′ (0) − Y ′ ( δ 4 ) > δ 2 } . W e then ha v e P λ,χ ( A 1 ,k ∪ A 2 ∪ A 3 ) ≤ ε ′ . Consequen tly , if we sho w that for k large enough, when A 1 ,k ∪ A 2 ∪ A 3 do es not o ccur w e hav e that for all x j ≤ y ≤ y 0 , R ′ j ( y ) − Λ R j,k ( y ) ∈ [0 , ε ] , this yields P λ,χ ( ∃ x j ≤ y ≤ y 0 , R ′ j ( y ) − Λ R j,k ( y ) ∈ [0 , ε ]) ≤ ε ′ when k is large enough, which pro v es the lemma. W e now assume that x j ≥ χ and A 1 ,k ∪ A 2 ∪ A 3 do es not o ccur, and prov e that for all x j ≤ y ≤ y 0 , R ′ j ( y ) − Λ R j,k ( y ) ∈ [0 , ε ] . Since A 1 ,k do es not o ccur, w e ha v e (3), th us if we denote Y k = inf { y ∈ [ x j , y 0 ] | Λ R j,k ( y ) = λ ( y ) } , w e ha v e Y k > x j and R ′ j ( x j ) − 3 · 2 − k ≤ Λ R j,k ( x j ) ≤ R ′ j ( x j ) . Then Y (0) ≥ Y k , and if y ∈ [ x j , Y k ] , w e ha v e R ′ j ( y ) − Λ R j,k ( y ) = R ′ j ( x j ) − Λ R j,k ( x j ) ∈ [0 , 3 · 2 − k ] ⊂ [0 , ε ] . If Y k = + ∞ , w e thus hav e R ′ j ( y ) − Λ R j,k ( y ) ∈ [0 , ε ] for all y ∈ [ x j , y 0 ] . If Y k < + ∞ , w e ha v e R ′ j ( y ) − Λ R j,k ( y ) ∈ [0 , 3 · 2 − k ] for y ∈ [ x j , Y k ] . In particular, R ′ j ( Y k ) ≤ λ ( Y k ) + 3 · 2 − k , so when k is large enough to ha v e 3 · 2 − k < δ 3 , we get Y ( δ 3 ) ≤ Y k . Moreo ver, since A 3 do es not occur, w e obtain Y (0) ≤ Y k + δ 2 , and for y ≥ Y (0) w e ha v e R ′ j ( y ) = Λ R j,k ( y ) = λ ( y ) . It remains to consider the case y ∈ [ Y k , Y k + δ 2 ] , y ≤ Y (0) . W e then ha v e Λ R j,k ( y ) = λ ( y ) ≤ R ′ j ( y ) . In addition, | Λ R j,k ( y ) − R ′ j ( y ) | = | λ ( y ) − R ′ j ( y ) | ≤ | λ ( y ) − λ ( Y (0)) | + | λ ( Y (0)) − R ′ j ( y ) | . F urthermore, b y the deﬁnition of δ 2 w e ha ve | λ ( y ) − λ ( Y (0)) | ≤ ε/ 2 , and | λ ( Y (0)) − R ′ j ( y ) | = | W j ( Y (0)) − W j ( y ) | ≤ ε/ 2 since A 2 do es not o ccur. Therefore | Λ R j,k ( y ) − R ′ j ( y ) | ≤ ε , hence R ′ j ( y ) − Λ R j,k ( y ) ∈ [0 , ε ] , which ends the pro of in the case x j ≥ χ . W e no w deal with the case x j < χ . W e assume A 1 ,k ∪ A 2 ∪ A 3 do es not occur and pro ve that for all x j ≤ y ≤ y 0 , R ′ j ( y ) − Λ R j,k ( y ) ∈ [0 , ε ] . W e begin b y studying R ′ j ( y ) − Λ R j,k ( y ) for y ∈ [ x, χ ] . W e denote Y = inf { y ∈ [ x, χ ] | R ′ j ( y ) = λ ( y ) } , Y k = inf { y ∈ [ x, χ ] | Λ R j,k ( y ) = λ ( y ) } (the inf being + ∞ if the set is empt y). Since A 1 ,k do es not occur, (3) yields Y k > x j and R ′ j ( x j ) − 3 · 2 − k ≤ Λ R j,k ( x j ) ≤ R ′ j ( x j ) . Then if y ∈ [ x j , Y k ] , w e ha v e R ′ j ( y ) − Λ R j,k ( y ) = R ′ j ( x j ) − Λ R j,k ( x j ) ∈ [0 , 3 · 2 − k ] ⊂ [0 , ε ] . This also implies Y k ≤ Y . F or Y k ≤ y ≤ Y (if Y k is ﬁnite), TRUE SELF-REPELLING MOTION ABOVE A GENERAL BARRIER 13 w e ha v e Λ R j,k ( y ) ≤ R ′ j ( y ) (indeed, if Λ R j,k ( y ) > R ′ j ( y ) then Λ R j,k ( y ) > λ ( y ) , and if y ′ = sup { y ′′ < y | Λ R j,k ( y ′′ ) = λ ( y ′′ ) } then Λ R j,k ( y ′ ) ≤ R ′ j ( y ′ ) , but Λ R j,k ( y ) − Λ R j,k ( y ′ ) = W j ( y ) − W j ( y ′ ) = R ′ j ( y ) − R ′ j ( y ′ ) whic h is a contradiction), and Λ R j,k ( y ) − Λ R j,k ( Y k ) ≥ W j ( y ) − W j ( Y k ) = R ′ j ( y ) − R ′ j ( Y k ) , hence 0 ≤ R ′ j ( y ) − Λ R j,k ( y ) ≤ R ′ j ( Y k ) − Λ R j,k ( Y k ) ≤ 3 · 2 − k , hence R ′ j ( y ) − Λ R j,k ( y ) ∈ [0 , ε ] . Finally (if Y is ﬁnite), λ ( Y ) ≤ Λ R j,k ( Y ) ≤ R ′ j ( Y ) = λ ( Y ) , so Λ R j,k ( Y ) = R ′ j ( Y ) thus for y ≥ Y we hav e Λ R j,k ( y ) = R ′ j ( y ) . Therefore R ′ j ( y ) − Λ R j,k ( y ) ∈ [0 , ε ] for all y ∈ [ x, χ ] . W e now consider x ∈ [ χ, y 0 ] (of course, this is not necessary if y 0 ≤ χ ). If λ ( χ ) = R ′ j ( χ ) = Λ R j,k ( χ ) , then R ′ j ( y ) = Λ R j,k ( y ) for all y ≥ χ . If λ ( χ ) = Λ R j,k ( χ ) < R ′ j ( χ ) , w e can obtain as in the case x j > χ that Y ( δ 3 ) ≤ Y k ≤ χ and Y (0) ∈ [ Y k , Y k + δ 2 ] . If y ∈ [ Y k , Y k + δ 2 ] , y ≤ Y (0) , we hav e R ′ j ( y ) − Λ R j,k ( y ) ≥ 0 (for the part of the interv al in [ x j , χ ] , it works as in the previous paragraph; for the other part b oth pro cesses start ev olving as W j , un til Λ R j,k meets λ , at which p oin t it is even low er). Moreov er, | R ′ j ( y ) − Λ R j,k ( y ) | ≤ | R ′ j ( y ) − λ ( y ) | ≤ ε , which can b e pro v en as in the case x j ≥ χ . Moreo ver, w e notice λ ( Y (0)) ≤ Λ R j,k ( Y (0)) ≤ R ′ j ( Y (0)) = λ ( Y (0)) , hence Λ R j,k ( y ) = R ′ j ( y ) for y ≥ Y (0) . This implies R ′ j ( y ) − Λ R j,k ( y ) ∈ [0 , ε ] for all y ∈ [ χ, y 0 ] . Finally , if λ ( χ ) < Λ R j,k ( χ ) ≤ R ′ j ( χ ) , w e notice that 0 ≤ R ′ j ( χ ) − Λ R j,k ( χ ) ≤ 3 · 2 − k , and we can complete the pro of with the same arguments as in the case x j ≥ χ , replacing the Y ( δ ) by the Y ′ ( δ ) . □ W e now pro v e the following lemma, which implies the ﬁnite-dimensional marginals of the Λ ′ j,k con v erge in probability to those of the C ′ j . Lemma 34. F or any j ∈ { 1 , ..., p } , y 0 > max 1 ≤ i ≤ r y i , ε > 0 , P λ,χ ( ∃ ℓ ∈ { 1 , ..., j } , ∃ x ℓ ≤ y ≤ y 0 , | C ′ ℓ ( y ) − Λ ′ ℓ,k ( y ) | > ε ) tends to 0 when k tends to + ∞ . Pr o of. W e pro v e the lemma by induction on j . The case j = 1 is given by Lemma 33. W e now set j ∈ { 2 , ..., p } and assume Lemma 34 holds for j − 1 . Let y 0 > max 1 ≤ i ≤ r y i , ε > 0 small, ε ′ > 0 , and let k ∈ N large. W e will deﬁne even ts with small probability so that if they do not o ccur then for all ℓ ∈ { 1 , ..., j } , x ℓ ≤ y ≤ y 0 , w e hav e | C ′ ℓ ( y ) − Λ ′ ℓ,k ( y ) | ≤ ε . In order to do that, we denote Y = inf { x ≥ x j | R ′ j ( y ) ∈ { C ′ 1 ( y ) , ..., C ′ j − 1 ( y ) }} (whic h is + ∞ if the set is empt y) the place at which R ′ j coalesces with one of the C ′ ℓ , ℓ ≤ j − 1 . If Y is ﬁnite, we denote L = { ℓ ∈ { 1 , ..., j − 1 } | R ′ j ( Y ) = C ′ ℓ ( Y ) } . W e c ho ose 0 < δ 1 ≤ ε so that, denoting A 1 = { Y < + ∞ , ∃ ℓ ∈ { 1 , ..., j − 1 } \ L , | C ′ ℓ ( Y ) − R ′ j ( Y ) | < δ 1 } , w e hav e P λ,χ ( A 1 ) ≤ ε ′ / 8 , so that, denoting A 2 = { λ ( Y ) < C ′ j ( Y ) < λ ( Y ) + δ 1 } , w e ha v e P λ,χ ( A 2 ) ≤ ε ′ / 8 , and so that, denoting A 3 = {∃ ℓ ∈ { 1 , ..., j − 1 } , Y ∈ ( x ℓ − δ 1 , x ℓ ) } , we ha ve P λ,χ ( A 3 ) ≤ ε ′ / 8 . In addition, λ, C ′ 1 , ..., C ′ j , R ′ j are uniformly contin uous on the resp ectiv e in terv als [ x j , y 0 + 1] , [ x 1 , y 0 + 1] , ..., [ x j , y 0 + 1] , [ x j , y 0 + 1] so we can c ho ose 0 < δ 2 < min(1 / 2 , δ 1 ) so that for y , y ′ ∈ [ x j , y 0 + 1] , if | y − y ′ | ≤ δ 2 then | λ ( y ) − λ ( y ′ ) | ≤ δ 1 / 6 , and denoting A 4 = {∃ ℓ ∈ { 1 , ..., j } , ∃ y , y ′ ∈ [ x ℓ , y 0 + 1] so that | y − y ′ | ≤ δ 2 and | C ′ ℓ ( y ) − C ′ ℓ ( y ′ ) | > δ 1 / 6 } ∪ {∃ y , y ′ ∈ [ x j , y 0 + 1] so that | y − y ′ | ≤ δ 2 and | R ′ j ( y ) − R ′ j ( y ′ ) | > δ 1 / 6 } , then P λ,χ ( A 4 ) ≤ ε ′ / 8 . W e no w deﬁne pro cesses ( W ℓ,k ( y )) y ≥ x ℓ , ℓ ∈ { 1 , ..., j − 1 } , where W ℓ,k will b e “the Bro wnian motion driving Λ ′ ℓ,k ”. W e deﬁne them b y induction as follo ws. W 1 ,k = W 1 . F or all ℓ ∈ { 2 , ..., j − 1 } , if Y ℓ,k = inf { y ≥ x ℓ | Λ R ℓ,k ( y ) ∈ { Λ ′ 1 ,k ( y ) , ..., Λ ′ ℓ − 1 ,k ( y ) }} and L ℓ,k = min { ℓ ′ ∈ { 1 , ..., ℓ − 1 } | Λ R ℓ,k ( Y ℓ,k ) = Λ ′ ℓ ′ ,k ( Y ℓ,k ) } , then W ℓ,k ( y ) = W ℓ ( y ) for x ℓ ≤ y ≤ Y ℓ,k and W ℓ,k ( y ) = W L ℓ,k ,k ( y ) − W L ℓ,k ,k ( Y ℓ,k ) + W ℓ ( Y ℓ,k ) for y ≥ Y ℓ,k . The W ℓ,k , ℓ ∈ { 1 , ..., j − 1 } are then Bro wnian motions. W e c ho ose 0 < δ 3 < δ 1 so that, calling A 5 = {∃ ℓ ∈ { 1 , ..., j − 1 } , ∃ y ∈ [max( x j , x ℓ ) , min( Y − δ 2 , y 0 )] , | R ′ j ( y ) − C ′ ℓ ( y ) | ≤ δ 3 } , then P λ,χ ( A 5 ) ≤ ε ′ / 8 , and so that, denoting A 6 ,k = {∃ ℓ ∈ { 1 , ..., j − 1 } , ∀ y ∈ [ Y , Y + δ 2 ] , W j ( y ) − W j ( Y ) − ( W ℓ,k ( y ) − W ℓ,k ( Y )) ≤ δ 3 } ∪ {∃ ℓ ∈ { 1 , ..., j − 1 } , ∀ y ∈ [ Y , Y + δ 2 ] , W j ( y ) − W j ( Y ) − ( W ℓ,k ( y ) − W ℓ,k ( Y )) ≥ − δ 3 } , then P λ,χ ( A 6 ,k ) ≤ ε ′ / 8 . Moreo v er, b y Lemma 33, if we denote A 7 ,k = {∃ ℓ ∈ { 1 , ..., j − 1 } , ∃ x ℓ ≤ y ≤ y 0 + 1 , R ′ ℓ ( y ) − Λ R ℓ,k ( y ) ∈ [0 , δ 3 / 6] } , then when 14 LAURE MARÊCHÉ k is large enough, P λ,χ ( A 7 ,k ) ≤ ε ′ / 8 . Finally , b y the induction hypothesis, if w e denote A 8 ,k = {∃ ℓ ∈ { 1 , ..., j − 1 } , ∃ x ℓ ≤ y ≤ y 0 + 1 , | C ′ ℓ ( y ) − Λ ′ ℓ,k ( y ) | > δ 3 / 6 } , then when k is large enough, P λ,χ ( A 8 ,k ) ≤ ε ′ / 8 . W e then hav e P λ,χ (( S 5 i =1 A i ) ∪ ( S 8 i =6 A i,k )) ≤ ε ′ when k is large enough. Therefore if we can show that when k is large enough and ( S 5 i =1 A i ) ∪ ( S 8 i =6 A i,k ) do es not o ccur, then for all ℓ ∈ { 1 , ..., j } , x ℓ ≤ y ≤ y 0 , | C ′ ℓ ( y ) − Λ ′ ℓ,k ( y ) | ≤ ε , we hav e P λ,χ ( ∃ ℓ ∈ { 1 , ..., j } , ∃ x ℓ ≤ y ≤ y 0 , | C ′ ℓ ( y ) − Λ ′ ℓ,k ( y ) | > ε ) ≤ ε ′ for k large enough, whic h pro v es the lemma for j and allo ws to complete the induction. W e now assume ( S 5 i =1 A i ) ∪ ( S 8 i =6 A i,k ) does not o ccur. F or an y ℓ ∈ { 1 , ..., j − 1 } , since A 5 do es not occur, for any y ∈ [max( x j , x ℓ ) , min( Y − δ 2 , y 0 )] w e ha v e | R ′ j ( y ) − C ′ ℓ ( y ) | > δ 3 . F urthermore, since A 7 ,k and A 8 ,k do not occur, we ha v e | R ′ j ( y ) − Λ R j,k ( y ) | ≤ δ 3 / 6 and | Λ ′ ℓ,k ( y ) − C ′ ℓ ( y ) | ≤ δ 3 / 6 , hence | Λ R j,k ( y ) − Λ ′ ℓ,k ( y ) | ≥ 2 δ 3 / 3 > 0 . Therefore Λ R j,k do es not coalesce with one of the Λ ′ ℓ,k b efore min( Y − δ 2 , y 0 ) , which implies Λ ′ j,k ( y ) = Λ R j,k ( y ) for y ∈ [ x j , min( Y − δ 2 , y 0 )] . Since A 7 ,k do es not o ccur, for all x j ≤ y ≤ min( Y − δ 2 , y 0 ) we deduce C ′ j ( y ) − Λ ′ j,k ( y ) ∈ [0 , ε ] (remem b er the deﬁnition of Y ). If Y ≥ y 0 + δ 2 , since A 8 ,k do es not o ccur, for all ℓ ∈ { 1 , ..., j } , x ℓ ≤ y ≤ y 0 , | C ′ ℓ ( y ) − Λ ′ ℓ,k ( y ) | ≤ ε , whic h is enough. W e now assume Y ≤ y 0 + δ 2 . It remains to pro v e that for all min( x j , Y − δ 2 ) ≤ y ≤ y 0 , | C ′ j ( y ) − Λ ′ j,k ( y ) | ≤ ε . Let ℓ ∈ { 1 , ..., j − 1 } \ L , we are going to sho w Λ R j,k do es not coalesce with one of the Λ ′ ℓ,k b efore Y + δ 2 . Since A 3 do es not o ccur, either x ℓ ≤ Y or x ℓ ≥ Y + δ 1 ≥ Y + δ 2 . F urthermore, if x ℓ ≤ Y , since A 1 do es not o ccur, | C ′ ℓ ( Y ) − C ′ j ( Y ) | ≥ δ 1 . Since A 4 o ccurs, if y ≥ max( x ℓ , x j ) and y ∈ [ Y − δ 2 , Y + δ 2 ] , | C ′ ℓ ( y ) − C ′ ℓ ( Y ) | ≤ δ 1 / 6 and | R ′ j ( y ) − R ′ j ( Y ) | ≤ δ 1 / 6 , hence | C ′ ℓ ( y ) − R ′ j ( y ) | ≥ 2 δ 1 / 3 . Moreo v er, A 7 ,k and A 8 ,k do not o ccur, so | R ′ j ( y ) − Λ R j,k ( y ) | ≤ δ 3 / 6 ≤ δ 1 / 6 and | Λ ′ ℓ,k ( y ) − C ′ ℓ ( y ) | ≤ δ 3 / 6 ≤ δ 1 / 6 , th us | Λ R j,k ( y ) − Λ ′ ℓ,k ( y ) | ≥ δ 1 / 3 > 0 , whic h means Λ R j,k do es not coalesce with one of the Λ ′ ℓ,k b efore Y + δ 2 . W e now study the interv al [ Y + δ 2 , y 0 + 1] . W e need to dinstinguish b et w een the cases C ′ j ( Y ) = λ ( Y ) and C ′ j ( Y ) > λ ( Y ) . W e begin by assuming C ′ j ( Y ) > λ ( Y ) . Since A 2 o ccurs, this implies C ′ j ( Y ) ≥ λ ( Y ) + δ 1 . Let y ∈ [ Y , Y + δ 2 ] . By the deﬁnition of δ 2 , λ ( y ) ≤ λ ( Y ) + δ 1 / 6 . Moreov er, since A 4 and A 8 ,k do not o ccur, for ℓ ∈ L , w e ha ve | Λ ′ ℓ,k ( y ) − C ′ j ( Y ) | ≤ | Λ ′ ℓ,k ( y ) − C ′ ℓ ( y ) | + | C ′ ℓ ( y ) − C ′ ℓ ( Y ) | ≤ δ 3 / 6 + δ 1 / 6 ≤ δ 1 / 3 , hence Λ ′ ℓ,k ( y ) > λ ( y ) . Similarly , since A 4 and A 7 ,k do not o ccur, Λ R j,k ( y ) > λ ( y ) . This implies Λ ′ ℓ,k ( y ) − Λ ′ ℓ,k ( Y ) = W ℓ,k ( y ) − W ℓ,k ( Y ) and Λ R j,k ( y ) − Λ R j,k ( Y ) = W j ( y ) − W j ( Y ) . In addition, the fact that A 6 ,k do es not o ccur yields the existence of y , y ′ ∈ [ Y , Y + δ 2 ] so that W j ( y ) − W j ( Y ) − ( W ℓ,k ( y ) − W ℓ,k ( Y )) > δ 3 and W j ( y ′ ) − W j ( Y ) − ( W ℓ,k ( y ′ ) − W ℓ,k ( Y )) < − δ 3 . Th us Λ R j,k ( y ) − Λ R j,k ( Y ) − (Λ ′ ℓ,k ( y ) − Λ ′ ℓ,k ( Y )) > δ 3 and Λ R j,k ( y ′ ) − Λ R j,k ( Y ) − (Λ ′ ℓ,k ( y ′ ) − Λ ′ ℓ,k ( Y )) < − δ 3 . F urthermore, | Λ ′ ℓ,k ( Y ) − Λ R j,k ( Y ) | ≤ | Λ ′ ℓ,k ( Y ) − C ′ ℓ ( Y ) | + | R ′ j ( Y ) − Λ R j,k ( Y ) | ≤ δ 3 / 3 since A 7 ,k , A 8 ,k do not o ccur. W e deduce Λ R j,k ( y ) − Λ ′ ℓ,k ( y ) ≥ 2 δ 3 / 3 > 0 and Λ R j,k ( y ′ ) − Λ ′ ℓ,k ( y ′ ) ≤ − 2 δ 3 / 3 < 0 . Therefore Λ R j,k has coalesced in [ Y − δ 2 , Y + δ 2 ] with some Λ ′ ℓ ′ ,k , ℓ ′ ∈ L . This implies that for any y ∈ [ Y + δ 2 , y 0 +1] , we hav e Λ ′ j,k ( y ) = Λ ′ ℓ ′ ,k ( y ) , with | Λ ′ ℓ ′ ,k ( y ) − C ′ ℓ ′ ( y ) | ≤ δ 3 / 6 since A 8 ,k do es not o ccur, and C ′ j ( y ) = C ′ ℓ ′ ( y ) , so | Λ ′ j,k ( y ) − C ′ j ( y ) | ≤ δ 3 / 6 ≤ ε . W e now consider the case C ′ j ( Y ) = λ ( Y ) . Then since A 7 ,k do es not o ccur, λ ( Y ) ≤ Λ R j,k ( Y ) ≤ R ′ j ( Y ) = λ ( Y ) , and if ℓ is the smallest elemen t of L , we ha v e Λ R ℓ,k ( Y ) ≤ R ′ ℓ ( Y ) . If we had C ′ ℓ ( Y )  = R ′ ℓ ( Y ) , then there would b e some ℓ ′ < ℓ so that C ′ ℓ ( Y ) = C ′ ℓ ′ ( Y ) , but this w ould con tradict the minimalit y of ℓ . Hence R ′ ℓ ( Y ) = C ′ ℓ ( Y ) = λ ( Y ) , whic h yields λ ( Y ) ≤ Λ R ℓ,k ( Y ) ≤ R ′ ℓ ( Y ) = λ ( Y ) , hence Λ R j,k ( Y ) = Λ R ℓ,k ( Y ) . In addition, if w e had Λ ′ ℓ,k ( Y )  = Λ R ℓ,k ( Y ) , there would b e some ℓ ′ < ℓ so that Λ ′ ℓ,k ( Y ) = Λ ′ ℓ ′ ,k ( Y ) . Since A 1 and A 8 ,k do not o ccur, this w ould imply | C ′ ℓ ′ ( Y ) − R ′ j ( Y ) | ≥ δ 1 and | C ′ ℓ ′ ( Y ) − Λ ′ ℓ ′ ,k ( Y ) | ≤ δ 3 / 6 , hence | Λ ′ ℓ ′ ,k ( Y ) − R ′ j ( Y ) | ≥ 5 δ 1 / 6 , th us | Λ ′ ℓ,k ( Y ) − C ′ ℓ ( Y ) | ≥ 5 δ 1 / 6 , whic h w ould con tradict the fact A 8 ,k do es not o ccur. Therefore we hav e Λ ′ ℓ,k ( Y ) = Λ R ℓ,k ( Y ) , hence Λ R j,k ( Y ) = Λ ′ ℓ,k ( Y ) . This implies TRUE SELF-REPELLING MOTION ABOVE A GENERAL BARRIER 15 Λ R j,k coalesces in [ Y − δ 2 , Y ] with some Λ ′ ℓ ′ ,k , ℓ ′ ∈ L . Consequen tly , for an y y ∈ [ Y , y 0 + 1] , w e ha v e Λ ′ j,k ( y ) = Λ ′ ℓ ′ ,k ( y ) , th us since A 8 ,k do es not o ccur, | Λ ′ j,k ( y ) − C ′ ℓ ′ ( y ) | ≤ δ 3 / 6 , hence | Λ ′ j,k ( y ) − C ′ j ( y ) | ≤ δ 3 / 6 ≤ ε for an y y ∈ [ Y , y 0 + 1] . It remains to consider the case y ∈ [ Y − δ 2 , Y + δ 2 ] , y ≥ x j . Then Λ ′ j,k ( y ) = Λ R j,k ( y ) or Λ ′ ℓ,k ( y ) with ℓ ∈ L . Since A 7 ,k and A 8 ,k do not o ccur, w e hav e | Λ ′ j,k ( y ) − R ′ j ( y ) | ≤ δ 3 / 6 in the ﬁrst case and | Λ ′ j,k ( y ) − C ′ ℓ ( y ) | ≤ δ 3 / 6 in the second. Since A 4 o ccurs, | R ′ j ( y ) − R ′ j ( Y ) | ≤ δ 1 / 6 , | C ′ ℓ ( y ) − C ′ ℓ ( Y ) | ≤ δ 1 / 6 and | C ′ j ( y ) − C ′ j ( Y ) | ≤ δ 1 / 6 , and remembering R ′ j ( Y ) = C ′ ℓ ( Y ) = C ′ j ( Y ) , w e obtain | Λ ′ j,k ( y ) − C ′ j ( Y ) | ≤ δ 3 / 6 + δ 1 / 6 ≤ δ 1 / 3 , hence | Λ ′ j,k ( y ) − C ′ j ( y ) | ≤ δ 1 / 3 + δ 1 / 6 = δ 1 / 2 ≤ ε . Consequen tly , for each y ∈ [ Y − δ 2 , Y + δ 2 ] , y ≥ x j , we ha v e | Λ ′ j,k ( y ) − C ′ j ( y ) | ≤ ε , which ends the proof of the lemma. □ Lemma 34 implies (Λ ′ j (1) ,k ( y 1 ) , ..., Λ ′ j ( r ) ,k ( y r )) conv erges in P λ,χ -probabilit y to ( C ′ j (1) ( y 1 ) , ..., C ′ j ( r ) ( y r )) when k tends to + ∞ , hence in distribution under P λ,χ . T o prov e that P λ,χ ( ∀ i ∈ { 1 , ..., r } , Λ ′ j ( i ) ,k ( y i ) ∈ ( a i , b i )) conv erges to P λ,χ ( ∀ i ∈ { 1 , ..., r } , C ′ j ( i ) ( y i ) ∈ ( a i , b i )) when k tends to + ∞ , whic h implies Proposition 32, b y the Portman teau Theorem it is enough to pro v e that for all i ∈ { 1 , ..., r } , we hav e P λ,χ ( C ′ j ( i ) ( y i ) = a i ) = P ( C ′ j ( i ) ( y i ) = b i ) = 0 , whic h is giv en by Lemma 28. □ 4.3. Lemma 8.2 of [32] . The pro of of Lemma 8.2 of [32] relies on an upper b ound on the v ariations of a RAB, which do es not hold in our case, since if a ( λ, χ ) -RAB hits the barrier, it can ﬂuctuate like the barrier itself, whic h may b e v ery wildly . Therefore we cannot use the argumen t of [32] as suc h. Ho w ev er, we can hav e an estimate on the ﬂuctuations of a ( λ, χ ) -RAB when it is “far from the barrier”, so w e mo dify the pro of in order to use the estimate only “far from the barrier”. F or any x ∈ R , w e denote M ′ ( x ) = { Λ ( ˜ x n , ˜ h n ) ( x ) | n ∈ N , ˜ x n < x, ( ˜ x n , ˜ h n ) ∈ R 2 λ } . The equiv alen t of Lemma 8.2 of [32] in our setting is the following lemma. Lemma 35. Almost sur ely, for al l x ∈ R , the set M ′ ( x ) is dense in [ λ ( x ) , + ∞ ) . Pr o of. It is enough to show that for an y dy adic num bers K > 0 , 0 < α < h , almost surely for all x ∈ ( − K, K ] we ha v e M ′ ( x ) ∩ ( λ ( x ) + h − α , λ ( x ) + h + α )  = ∅ . Let K > 0 , 0 < α < h b e dyadic n um b ers. λ is a con tinuous function, hence uniformly con tin uous on [ − K, K ] , so there exists some ﬁnite random k 0 so that when k ≥ k 0 , for all x, y ∈ [ − K, K ] so that | x − y | ≤ 2 − k K w e hav e | λ ( x ) − λ ( y ) | ≤ α/ 3 . F or any j ∈ {− 2 k , ..., 2 k − 1 } , w e choose a k,j a random dy adic n um b er so that | a k,j − ( λ ( j 2 − k K ) + h ) | ≤ α/ 3 . If for all j ∈ {− 2 k , ..., 2 k − 1 } , for all x ∈ [ j 2 − k K, ( j + 1)2 − k K ] we ha v e | Λ ( j 2 − k K,a k,j ) ( x ) − a k,j | < α/ 3 , then for all x ∈ ( − K, K ] , for j ∈ {− 2 k , ..., 2 k − 1 } such x ∈ ( j 2 − k K, ( j + 1)2 − k K ] , when k ≥ k 0 w e hav e | Λ ( j 2 − k K,a k,j ) ( x ) − ( λ ( x ) + h ) | ≤ | Λ ( j 2 − k K,a k,j ) ( x ) − a k,j | + | a k,j − ( λ ( j 2 − k K ) + h ) | + | λ ( j 2 − k K ) − λ ( x ) | < α, hence M ′ ( x ) ∩ ( λ ( x ) + h − α, λ ( x ) + h + α )  = ∅ . Moreov er, thanks to Lemma 27, for any j ∈ {− 2 k , ..., 2 k − 1 } , we ha ve P  k ≥ k 0 , ∃ x ∈ [ j 2 − k K, ( j + 1)2 − k K ] , | Λ ( j 2 − k K,a k,j ) ( x ) − a k,j | ≥ α/ 3  ≤ X ℓ ∈ D E  1 { k ≥ k 0 ,a k,j = ℓ } P λ,χ  ∃ x ∈ [ j 2 − k K, ( j + 1)2 − k K ] , | Λ ( j 2 − k K,ℓ ) ( x ) − ℓ | ≥ α/ 3  ≤ X ℓ ∈ D E 1 { k ≥ k 0 ,a k,j = ℓ } 6 √ 2 v K 2 − k α √ π exp  − α 2 18 v K 2 − k  ! ≤ 6 √ 2 v K 2 − k α √ π exp  − α 2 18 v K 2 − k  . 16 LAURE MARÊCHÉ Therefore P ( k ≥ k 0 , ∃ j ∈ {− 2 k , ..., 2 k − 1 } , ∃ x ∈ [ j 2 − k K, ( j + 1)2 − k K ] , | Λ ( j 2 − k K,a k,j ) ( x ) − a k,j | ≥ α/ 3) ≤ 12 · 2 k/ 2 √ 2 v K α √ π exp( − α 2 18 v K 2 k ) , which tends to 0 when k tends to + ∞ . This implies P ( k ≥ k 0 , ∀ j ∈ {− 2 k , ..., 2 k − 1 } , ∀ x ∈ [ j 2 − k K, ( j + 1)2 − k K ] , | Λ ( j 2 − k K,a k,j ) ( x ) − a k,j | < α/ 3) tends to 1 when k tends to + ∞ . W e deduce that almost surely , for all x ∈ ( K , − K ] we ha v e M ′ ( x ) ∩ ( λ ( x ) + h − α, λ ( x ) + h + α )  = ∅ , whic h ends the proof of the lemma. □ 4.4. Equation (8.49) of [32] . Lik e Lemma 8.1 of [32], their Equation (8.49) relies on an estimate of the probability t w o RABs coalesce, which w e do not ha v e in our setting, so w e hav e to give another pro of. Let q < q ′ b e tw o elemen ts of D . F or any K ∈ N ∗ , p ∈ N ∗ , we denote N K p = card( { Λ ( q , 2 − p j ) ( q ′ ) | j ∈ {− 2 p K, ..., 2 p K } , 2 − p j > λ ( q ) } ) . T o replace Equation (8.49) of [32], w e need an upp er b ound on E ( N K p ) which do es not dep end on p , whic h will b e the follo wing lemma. Lemma 36. E ( N K p ) ≤ 2 + 2 K √ π v ( q ′ − q ) . Pr o of. The proof in [32] relies on a small upper b ound on P (Λ ( q , 2 − p ( j − 1)) ( q ′ ) < Λ ( q , 2 − p j ) ( q ′ )) , whic h we do not ha v e in our setting b ecause of the p ossibly wild ﬂuctuations of λ . Ho wev er, we will b e able to b ound P ( λ ( q ′ ) < Λ ( q , 2 − p ( j − 1)) ( q ′ ) < Λ ( q , 2 − p j ) ( q ′ )) , as a w a y from λ the pro cesses Λ ( q , 2 − p ( j − 1)) , Λ ( q , 2 − p j ) b eha ve lik e Bro wnian motions. T o make this w eaker b ound suﬃcien t, our idea will b e to notice that only the low est Λ ( q , 2 − p j ) ( q ′ ) can b e equal to λ ( q ′ ) . W e consider { Λ ( q , 2 − p j ) ( q ′ ) | j ∈ {− 2 p K, ..., 2 p K } , 2 − p j > λ ( q ) } . It may contain λ ( q ′ ) , the smallest Λ ( q , 2 − p j ) ( q ′ ) strictly larger than λ ( q ′ ) , and all the Λ ( q , 2 − p j ′ ) ( q ′ ) larger than this Λ ( q , 2 − p j ) ( q ′ ) . Therefore E ( N K p ) ≤ 2 + 2 p K X j = − 2 p K +2 P (2 − p ( j − 1) > λ ( q ) , λ ( q ′ ) < Λ ( q , 2 − p ( j − 1)) ( q ′ ) < Λ ( q , 2 − p j ) ( q ′ )) . Moreo v er, for an y j ∈ {− 2 p K + 2 , ..., 2 p K } , b y Theorem 6 the pro cesses Λ ( q , 2 − p ( j − 1)) and Λ ( q , 2 − p j ) form a ( λ, χ ) - FICRAB starting from ( q , 2 − p ( j − 1)) and ( q , 2 − p j ) . Let x ∈ R , h < h ′ , let ( W y ) y ≥ x , ( W ′ y ) y ≥ x b e tw o indep enden t Bro wnian motions with W x = h , W ′ x = h ′ , let ( R y ) y ≥ x , ( R ′ y ) y ≥ x b e the ( λ, χ ) -RABs driv en by ( W y ) y ≥ x , ( W ′ y ) y ≥ x starting from ( x, h ) , ( x, h ′ ) , and (( C y ) y ≥ x , ( C ′ y ) y ≥ x ) b e the ( λ, χ ) -FICRAB constructed from ( R y ) y ≥ x , ( R ′ y ) y ≥ x . In order to pro v e the lemma, it is enough to prov e that for an y y > x we ha v e P ( h > λ ( x ) , λ ( y ) < C y < C ′ y ) ≤ h ′ − h √ π v ( y − x ) . Indeed, this implies E ( N K p ) ≤ 2 + 2 p K X j = − 2 p K +2 2 − p p π v ( q ′ − q ) ≤ 2 + 2 · 2 p K 2 − p p π v ( q ′ − q ) = 2 + 2 K p π v ( q ′ − q ) . W e no w prov e that if h > λ ( x ) and λ ( y ) < C y < C ′ y then for all z ∈ [ x, y ] , W z < W ′ z . W e assume by con tradiction that there exists z 0 ∈ [ x, y ] so that W z 0 ≥ W ′ z 0 . There are several cases. If there exists z ∈ [ x, y ] so that W ′ z = λ ( z ) , then if ¯ z is the inf of these z we ha v e R ′ ¯ z = W ′ ¯ z = λ ( ¯ z ) . W e then hav e R ′ ¯ z ≤ R ¯ z while R ′ x > R x , so there exists z ∈ [ x, ¯ z ] so that R z = R ′ z , hence C y = C ′ y , whic h gives a contradiction. W e no w assume that for all z ∈ [ x, y ] w e hav e W ′ z > λ ( z ) , whic h implies for all z ∈ [ x, y ] w e hav e R ′ z = W ′ z . In addition, if R y = λ ( y ) then C y = λ ( y ) which would b e a contradiction, so we ma y assume R y > λ ( y ) . By the construction of ( R y ) y ≥ x , it implies that for all z ∈ [ x, y ] we ha v e R z ≥ W z . In particular, R z 0 ≥ W z 0 ≥ W ′ z 0 = R ′ z 0 , while R ′ x > R x , hence there exists z ∈ [ x, z 0 ] so that R z = R ′ z , hence C y = C ′ y whic h gives a contradiction. Consequen tly , if h > λ ( x ) and λ ( y ) < C y < C ′ y then for all z ∈ [ x, y ] , TRUE SELF-REPELLING MOTION ABOVE A GENERAL BARRIER 17 W z < W ′ z . This yields P ( h > λ ( x ) , λ ( y ) < C y < C ′ y ) ≤ P ( ∀ z ∈ [ x, y ] , W z < W ′ z ) ≤ h ′ − h √ π v ( y − x ) b y Lemma 25, which ends the pro of of the lemma. □ 5. Ba ckw ard lines: proof of Theorem 11 T o pro v e Theorem 11, whic h states (Λ ∗ ( − x,h ) ( − . )) ( x,h ) ∈ R 2 is a system of forward lines ab ov e ( λ ( − . ) , − χ ) , we sho w (Λ ∗ ( − x,h ) ( − . )) ( x,h ) ∈ R 2 satisﬁes the prop erties of Theorem 6, which characterize the law of a system of forward lines. The last three prop erties of Theorem 6 can b e pro ven as in [32] (see their Lemma 9.1); the diﬃcult part is to pro ve that for any p ∈ N ∗ , if ( x 1 , h 1 ) , ..., ( x p , h p ) ∈ R 2 , then (Λ ∗ ( − x 1 ,h 1 ) ( − . ) , ..., Λ ∗ ( − x p ,h p ) ( − . )) is a ( λ ( − . ) , − χ ) -FICRAB. As in [32], we ﬁrst pro v e that conditionally to ( λ, χ ) , the backw ard lines are Marko v pro cesses in Section 5.1, then determine their transition probabilities in Section 5.2, contin ue b y sho wing they are indep enden t as long as they sta y apart in Section 5.3, and ﬁnally prov e that they merge when they meet in Section 5.4. Eac h of these p oints needs mo diﬁcations with resp ect to [32]. In this section, we w ork conditionally to ( λ, χ ) ev en when it is not men tioned. 5.1. Mark o v property of the bac kw ard lines giv en ( λ, χ ) . The pro of of the Marko v prop erty of the Λ ∗ ( x,h ) in [32] relies on the fact that for x 0 < x , at the left of x 0 w e ha v e almost surely Λ ∗ ( x,h ) = Λ ∗ ( x 0 , Λ ∗ ( x,h ) ( x 0 )) , whic h do es not dep end on what happ ens at the righ t of x 0 . How ever, this do es not work if Λ ∗ ( x,h ) ( x 0 ) = λ ( x 0 ) , whic h may happ en with positive probability with our more general λ , thus we need additional argumen ts, whic h are given in the follo wing prop osition. It is in its proof that we need the assumption ( λ, χ ) nice. Prop osition 37. F or any ( x, h ) ∈ R 2 λ , x 0 < x , we have the fol lowing. • If x 0 ≤ χ , almost sur ely if Λ ∗ ( x,h ) ( x 0 ) = λ ( x 0 ) then for al l y ≤ x 0 we have Λ ∗ ( x,h ) ( y ) = λ ( y ) . • If x 0 > χ , if Λ ∗ ( x,h ) ( x 0 ) = λ ( x 0 ) then for al l y ≤ x 0 we have Λ ∗ ( x,h ) ( y ) = sup { Λ ( ˜ x n , ˜ h n ) ( y ) | ˜ x n < y , Λ ( ˜ x n , ˜ h n ) ( x 0 ) = λ ( x 0 ) } (wher e the sup is λ ( y ) if the set is empty). If we assume Proposition 37, w e can prov e the Mark ov property of the backw ard lines with arguments similar to those of [32]. Indeed, one can pro v e as in Lemma 9.2 of [32] that for an y ( x, h ) ∈ R 2 λ , x 0 ≤ x , almost surely if Λ ∗ ( x,h ) ( x 0 ) > λ ( x 0 ) then for all y ≤ x 0 w e ha v e Λ ∗ ( x 0 , Λ ∗ ( x,h ) ( x 0 )) ( y ) = Λ ∗ ( x,h ) ( y ) . Moreov er, one can pro v e as they do just before that lemma that the families { Λ ( x,h ) ( y ) , x ≤ y ≤ x 0 , h > λ ( x ) } ∪ { Λ ∗ ( x,h ) ( y ) , y ≤ x ≤ x 0 , h > λ ( x ) } and { Λ ∗ ( x,h ) ( y ) , x 0 ≤ y ≤ x, h > λ ( x ) } are indep endent. This implies that for any ( x 1 , h 1 ) , ..., ( x p , h p ) ∈ R 2 λ , (Λ ∗ ( x 1 ,h 1 ) ( − . ) , ..., Λ ∗ ( x p ,h p ) ( − . )) is a Mark o v pro cess. Pr o of of Pr op osition 37. W e will use diﬀeren t argumen ts dep ending on whether x 0 > χ , x 0 = χ or x 0 < χ . W e begin with the case x 0 > χ . W e assume Λ ∗ ( x,h ) ( x 0 ) = λ ( x 0 ) . Let y ≤ x 0 . Then if m ∈ N is so that ˜ x m < y and Λ ( ˜ x m , ˜ h m ) ( x ) < h , we ha ve ˜ x m < x 0 and Λ ( ˜ x m , ˜ h m ) ( x ) < h hence Λ ( ˜ x m , ˜ h m ) ( x 0 ) = λ ( x 0 ) . Con v ersely , if m ∈ N is so that ˜ x m < y and Λ ( ˜ x m , ˜ h m ) ( x 0 ) = λ ( x 0 ) , then Λ ( ˜ x m , ˜ h m ) ( x ) = λ ( x ) < h . W e deduce { m ∈ N | ˜ x m < y , Λ ( ˜ x m , ˜ h m ) ( x ) < h } = { m ∈ N | ˜ x m < y , Λ ( ˜ x m , ˜ h m ) ( x 0 ) = λ ( x 0 ) } , hence Λ ∗ ( x,h ) ( y ) = sup { Λ ( ˜ x n , ˜ h n ) ( y ) | ˜ x n < y , Λ ( ˜ x n , ˜ h n ) ( x 0 ) = λ ( x 0 ) } . W e now consider the case x 0 = χ . It is enough to sho w that almost surely , for all n ∈ N so that ˜ x n < χ , ˜ h n > λ ( ˜ x n ) , w e hav e Λ ( ˜ x n , ˜ h n ) ( χ ) > λ ( χ ) . Since ( λ, χ ) is nice, this comes from Lemma 29. W e no w deal with the case x 0 < χ . W e will use the follo wing lemma, prov en later. 18 LAURE MARÊCHÉ Lemma 38. F or any x 0 < χ , x > x 0 , n ∈ N so that ˜ x n < x 0 , almost sur ely if Λ ( ˜ x n , ˜ h n ) ( x 0 ) = λ ( x 0 ) , ther e exists h ′ > λ ( x 0 ) so that Λ ( x 0 ,h ′ ) ( x ) = Λ ( ˜ x n , ˜ h n ) ( x ) . W e no w assume that for any n ∈ N so that ˜ x n < x 0 , if Λ ( ˜ x n , ˜ h n ) ( x 0 ) = λ ( x 0 ) , there exists h ′ > λ ( x 0 ) so that Λ ( x 0 ,h ′ ) ( x ) = Λ ( ˜ x n , ˜ h n ) ( x ) , whic h happ ens almost surely by Lemma 38. W e recall Lemma 35: almost surely for eac h y ∈ R , the set M ′ ( y ) = { Λ ( ˜ x m , ˜ h m ) ( y ) | m ∈ N , ˜ x m < y , ( ˜ x m , ˜ h m ) ∈ R 2 λ } is dense in [ λ ( y ) , + ∞ ) (since it holds for any c hoice of ( λ, χ ) , it holds under our conditioning). T ogether these almost sure ev ents imply that, for all n ∈ N so that ˜ x n < x 0 , if Λ ( ˜ x n , ˜ h n ) ( x 0 ) = λ ( x 0 ) , there exists m ∈ N so that ˜ x m < x 0 , Λ ( ˜ x m , ˜ h m ) ( x 0 ) > λ ( x 0 ) and Λ ( ˜ x m , ˜ h m ) ( x ) = Λ ( ˜ x n , ˜ h n ) ( x ) . Consequen tly , if there exists n ∈ N so that ˜ x n < x 0 , Λ ( ˜ x n , ˜ h n ) ( x ) < h and Λ ( ˜ x n , ˜ h n ) ( x 0 ) = λ ( x 0 ) , then there exists m ∈ N so that ˜ x m < x 0 , Λ ( ˜ x m , ˜ h m ) ( x 0 ) > λ ( x 0 ) and Λ ( ˜ x m , ˜ h m ) ( x ) < h , hence Λ ∗ ( x,h ) ( x 0 ) > λ ( x 0 ) . Therefore if Λ ∗ ( x,h ) ( x 0 ) = λ ( x 0 ) , there is no n ∈ N so that ˜ x n < x 0 , Λ ( ˜ x n , ˜ h n ) ( x ) < h and Λ ( ˜ x n , ˜ h n ) ( x 0 ) = λ ( x 0 ) , thus there is no n ∈ N so that ˜ x n < x 0 and Λ ( ˜ x n , ˜ h n ) ( x ) < h . This implies Λ ∗ ( x,h ) ( y ) = λ ( y ) for all y ≤ x 0 , whic h ends the pro of of Prop osition 37 p ending the proof of Lemma 38. □ Pr o of of L emma 38. Let x 0 < χ , x > x 0 , n ∈ N so that ˜ x n < x 0 . If x > χ , w e replace x b y χ , since if there is coalescence before χ there is coalescence before x . The idea of the proof is that if Λ ( x 0 ,h ′ ) and Λ ( ˜ x n , ˜ h n ) w ere Bro wnian motions with h ′ close to Λ ( ˜ x n , ˜ h n ) ( x 0 ) they w ould be v ery lik ely to meet, and since we work in the p ortion of the space where they are reﬂected instead, the barrier will “push up the pro cess which is b elow”, increasing even more the lik elihoo d of the tw o pro cesses meeting. F or an y k ∈ N ∗ , we remem b er (Λ ( ˜ x n , ˜ h n ) , Λ ( x 0 ,λ ( x 0 )+2 − k ) ) is a ( λ, χ ) -FICRAB starting from ( ˜ x n , ˜ h n ) , ( x 0 , λ ( x 0 ) + 2 − k ) . Let ( ˜ W z ) z ≥ ˜ x n , ( W z ) z ≥ x 0 b e t w o independent Bro wnian motions. W e deﬁne ( ˜ R ( y )) y ≥ ˜ x n as the ( λ, χ ) -RAB starting from ( ˜ x n , ˜ h n ) and driv en b y ( ˜ W z ) z ≥ ˜ x n , and for an y k ≥ 1 , we deﬁne ( R k ( y )) y ≥ ˜ x n as the ( λ, χ ) -RAB starting from ( x 0 , λ ( x 0 ) + 2 − k ) and driven b y ( W z ) z ≥ x 0 . Denoting A k = { ˜ R ( x 0 ) = λ ( x 0 ) , ∀ y ∈ [ x 0 , x ] , ˜ R ( y ) < R k ( y ) } , it is enough to pro v e P λ,χ ( A k ) tends to 0 when k tends to + ∞ . W e now consider the ev en t A ′ k = {∀ y ∈ [ x 0 , x ] , ˜ W y − ˜ W x 0 < W y − W x 0 + 2 − k } . W e are going to prov e that if A ′ k do es not o ccur and ˜ R ( x 0 ) = λ ( x 0 ) , then there exists y ∈ [ x 0 , x ] so that ˜ R ( y ) ≥ R k ( y ) , which implies A k ⊂ A ′ k . W e assume A ′ k do es not o ccur and ˜ R ( x 0 ) = λ ( x 0 ) . If there exists y ∈ [ x 0 , x ] so that R k ( y ) = λ ( y ) , then ˜ R ( y ) ≥ R k ( y ) . W e now consider the case in whic h R k ( y ) > λ ( y ) for all y ∈ [ x 0 , x ] , which implies R k ( y ) = λ ( x 0 ) + 2 − k + W y − W x 0 for all y ∈ [ x 0 , x ] . Since A ′ k do es not occur, there exists y ∈ [ x 0 , x ] so that ˜ W y − ˜ W x 0 ≥ W y − W x 0 + 2 − k . In addition, ˜ R ( y ) ≥ ˜ R ( x 0 ) + ˜ W y − ˜ W x 0 ≥ λ ( x 0 ) + W y − W x 0 + 2 − k = R k ( y ) . Consequently A k ⊂ A ′ k . In addition, Lemma 25 yields P λ,χ ( A ′ k ) ≤ 2 − k √ π v ( x − x 0 ) . W e deduce P λ,χ ( A k ) tends to 0 when k tends to + ∞ , whic h ends the pro of of Lemma 38. □ 5.2. T ransition probabilities of the backw ard lines. In this section w e pro v e that the backw ard lines hav e the righ t transition probabilities, which is Prop osition 39. The extension to our setting of the study of the transition probabilities made in [32] has tw o ma jor obstacles. The ﬁrst one is that it relies on a relationship they establish b et ween the distributions of reﬂected and absorb ed Bro wnian motions (their Equation (9.25)), which crucially depends on the fact their barrier is the abscissa axis. W e th us hav e to replace their equation b y Lemma 40. Moreov er, the argumen ts of [32] do not allow to deal with the case Λ ∗ ( x,h ) ( x ′ ) = λ ( x ′ ) . This w as not a problem for them since they had P (Λ ∗ ( x,h ) ( x ′ ) = λ ( x ′ )) = 0 , but with our more general barriers this even t ma y ha ve a p ositive probabilit y , hence w e hav e to treat this case separately . TRUE SELF-REPELLING MOTION ABOVE A GENERAL BARRIER 19 Prop osition 39. L et ( x, h ) ∈ R 2 λ , x ′′ < x ′ ≤ x , h ′ ≥ λ ( x ′ ) , if Λ ∗ ( x,h ) ( x ′ ) = h ′ , then Λ ∗ ( x,h ) ( x ′′ ) has the law of the mar ginal at − x ′′ of a ( λ ( − . ) , − χ ) -RAB starting at ( − x ′ , h ′ ) , and a b arrier-starting one if h ′ = λ ( x ′ ) (se e R emark 3). Pr o of. W e can consider only the cases x ′ ≤ χ and x ′′ ≥ χ , as knowing the transition probabilities from x ′ to χ and from χ to x ′′ yields the transition probability from x ′ to x ′′ . W e ﬁrst assume h ′ > λ ( x ′ ) . The result follo ws from the pro of on Page 433 of [32] if we replace their Equation (9.25) b y the following lemma. Lemma 40. L et ( W ← − y ) − x ′ ≤ y ≤− x ′′ b e a ( λ ( − . ) , − χ ) -RAB starting fr om ( − x ′ , h ′ ) 1 , then for any h ′′ > λ ( x ′′ ) we have P λ,χ (Λ ( x ′′ ,h ′′ ) ( x ′ ) > h ′ ) = P λ,χ ( W ← x ′′ < h ′′ ) . Pr o of. Let ( W y ) y ∈ [ x ′′ ,x ′ ] b e a Brownian motion with W x ′′ = 0 . W e deﬁne ( W ← − y ) − x ′ ≤ y ≤− x ′′ as the ( λ ( − . ) , − χ ) -RAB starting from ( − x ′ , h ′ ) and driven by ( W − y − W x ′ ) − x ′ ≤ y ≤− x ′′ . W e also deﬁne ( W → y ) x ′′ ≤ y ≤ x ′ as the ( λ, χ ) -RAB starting from ( x ′′ , h ′′ ) and driv en b y ( W y ) y ∈ [ x ′′ ,x ′ ] . ( W → y ) x ′′ ≤ y ≤ x ′ has the same distribution as Λ ( x ′′ ,h ′′ ) | [ x ′′ ,x ′ ] , hence it is enough to prov e P λ,χ ( W → x ′ > h ′ ) = P λ,χ ( W ← x ′′ < h ′′ ) . W e b egin by assuming x ′′ ≥ χ . Then ( W → y ) x ′′ ≤ y ≤ x ′ is a Brownian motion absorb ed on λ , which may meet λ or not. W e ﬁrst deal with the situation in which ( W → y ) x ′′ ≤ y ≤ x ′ do es not meet λ . Then, for all y ∈ [ x ′′ , x ′ ] w e hav e h ′′ + W y > λ ( y ) . W e ﬁrst assume W → x ′ > h ′ , hence h ′′ + W x ′ > h ′ . Then since ( W ← − y ) − x ′ ≤ y ≤− x ′′ starts at ( − x ′ , h ′ ) , for an y y ∈ [ − x ′ , − x ′′ ] , w e ha v e W ← − y < h ′′ + W − y (indeed the t w o processes W ← − . and h ′′ + W − . cannot cross), th us W ← x ′′ < h ′′ . W e now assume W → x ′ ≤ h ′ , whic h means h ′′ + W x ′ ≤ h ′ . Then for all y ∈ [ − x ′ , − x ′′ ] , w e hav e h ′ + W − y − W x ′ ≥ h ′′ + W − y > λ ( y ) , hence ( W ← y ) x ′′ ≤ y ≤ x ′ do es not meet λ , hence W ← x ′′ = h ′ + W x ′′ − W x ′ ≥ h ′′ . Therefore, if ( W → y ) x ′′ ≤ y ≤ x ′ do es not meet λ , then W → x ′ > h ′ is equiv alen t to W ← x ′′ < h ′′ (in the same w a y , W → x ′ ≥ h ′ is equiv alent to W ← x ′′ ≤ h ′′ ). W e no w deal with the situation in whic h ( W ← y ) x ′′ ≤ y ≤ x ′ meets λ . Since h ′ > λ ( x ′ ) , we ha ve W → x ′ < h ′ . F urthermore, let Y = inf { y ≥ x ′ | h ′′ + W y = λ ( y ) } the place where ( W ← y ) x ′′ ≤ y ≤ x ′ meets λ . F or y ∈ [ x ′′ , Y ] , w e ha v e W ← y − W ← Y ≥ W y − W Y = W → y − W → Y = W → y − λ ( Y ) . Since W ← Y ≥ λ ( Y ) , w e get W ← x ′′ ≥ W → x ′′ = h ′′ , th us W ← x ′′ ≥ h ′′ . W e deduce { W → x ′ > h ′ } = { W ← x ′′ < h ′′ } , hence P λ,χ ( W → x ′ > h ′ ) = P λ,χ ( W ← x ′′ < h ′′ ) . W e now consider the case x ′ ≤ χ . Then the roles of ( W → y ) x ′′ ≤ y ≤ x ′ and ( W ← y ) x ′′ ≤ y ≤ x ′ are rev ersed with resp ect to the previous case: ( W → y ) x ′′ ≤ y ≤ x ′ is reﬂected and ( W ← y ) x ′′ ≤ y ≤ x ′ is absorb ed. Therefore the previous reasoning yields { W → x ′ > h ′ } ⊂ { W ← x ′′ < h ′′ } and { W ← x ′′ < h ′′ } ⊂ { W → x ′ ≥ h ′ } . Moreov er, Lemma 28 implies P λ,χ ( W → x ′ = h ′ ) = 0 , hence P λ,χ ( W → x ′ > h ′ ) = P λ,χ ( W ← x ′′ < h ′′ ) . □ W e also need to deal with the case h ′ = λ ( x ′ ) . Then only x ′ < x is interesting. If x ′ ≤ χ , Prop osition 37 yields that almost surely , if Λ ∗ ( x,h ) ( x ′ ) = λ ( x ′ ) then Λ ∗ ( x,h ) ( y ) = λ ( y ) for all y ≤ x ′ , which is enough. W e now assume x ′′ ≥ χ and study the distribution of Λ ∗ ( x,h ) ( x ′′ ) if Λ ∗ ( x,h ) ( x ′ ) = λ ( x ′ ) . W e notice that if Λ ∗ ( x,h ) ( x ′ ) = λ ( x ′ ) , for an y ˆ x < x ′ , ˆ h > λ ( ˆ x ) , either Λ ( ˆ x, ˆ h ) ( x ′ ) = λ ( x ′ ) or Λ ( ˆ x, ˆ h ) ( x ) ≥ h . Consequently , for h ′′ > λ ( x ′′ ) , an approac h similar to that of Equations (9.22) and (9.23) of [32] yields P λ,χ (Λ ∗ ( x,h ) ( x ′′ ) > h ′′ | Λ ∗ ( x,h ) ( x ′ ) = λ ( x ′ )) ≥ P λ,χ (Λ ( x ′′ ,h ′′ ) ( x ) < h | Λ ∗ ( x,h ) ( x ′ ) = λ ( x ′ )) = P λ,χ (Λ ( x ′′ ,h ′′ ) ( x ′ ) = λ ( x ′ )) , P λ,χ (Λ ∗ ( x,h ) ( x ′′ ) ≤ h ′′ | Λ ∗ ( x,h ) ( x ′ ) = λ ( x ′ )) ≥ P λ,χ (Λ ( x ′′ ,h ′′ ) ( x ) ≥ h | Λ ∗ ( x,h ) ( x ′ ) = λ ( x ′ )) = P λ,χ (Λ ( x ′′ ,h ′′ ) ( x ′ )  = λ ( x ′ )) , therefore P λ,χ (Λ ∗ ( x,h ) ( x ′′ ) > h ′′ | Λ ∗ ( x,h ) ( x ′ ) = λ ( x ′ )) = P λ,χ (Λ ( x ′′ ,h ′′ ) ( x ′ ) = λ ( x ′ )) . 1 Or rather its restriction to [ − x ′ , − x ′′ ] . 20 LAURE MARÊCHÉ In addition, we notice that P λ,χ (Λ ( x ′′ ,h ′′ ) ( x ′ ) = λ ( x ′ )) is the probability a ( λ, χ ) -RAB starting from ( x ′′ , h ′′ ) is absorb ed b efore x ′ . Let ( W y ) y ∈ R b e a Bro wnian motion with W x ′′ = 0 , we th us ha v e P λ,χ (Λ ( x ′′ ,h ′′ ) ( x ′ ) = λ ( x ′ )) = P λ,χ ( ∃ y ∈ [ x ′′ , x ′ ] , h ′′ + W y ≤ λ ( y )) = P λ,χ (sup x ′′ ≤ y ≤ x ′ ( λ ( y ) − W y ) ≥ h ′′ ) = P λ,χ ( W x ′′ − W x ′ +sup x ′′ ≤ y ≤ x ′ ( λ ( y ) − ( W y − W x ′ )) ≥ h ′′ ) . Denoting ( R y ) y ≥− x ′ the barrier-starting ( λ ( − . ) , − χ ) -RAB starting from ( − x ′ , λ ( x ′ )) and driv en b y ( W − y − W x ′ ) y ≥− x ′ , w e deduce P λ,χ (Λ ( x ′′ ,h ′′ ) ( x ′ ) = λ ( x ′ )) = P λ,χ ( R x ′′ ≥ h ′′ ) = P λ,χ ( R x ′′ > h ′′ ) b y Lemma 28, whic h ends the pro of of Prop osition 39. □ 5.3. Indep endence of the bac kw ard lines un til they meet. W e need to prov e that if ( x, h 1 ) , ( x, h 2 ) ∈ R 2 λ , then Λ ∗ ( x,h 1 ) ( − . ) and Λ ∗ ( x,h 2 ) ( − . ) are indep enden t un til they meet. In tuitiv ely , the idea of the pro of is that the pro cesses are independent as long as they liv e in disjoin t boxes of R 2 , so we enclose their tra jectories in a sequence of disjoin t b o xes un til they get to o close to each other. Unfortunately , b ecause of our more general barriers, w e must change the deﬁnition of the b oxes given in [32]. Indeed, the b o xes are constructed so that when Λ ∗ ( x,h i ) en ters one of them, it is v ery unlikely to ﬂuctuate enough to go ab o ve the top of the box or b elo w its b ottom. In [32], they hav e uniform estimates on the ﬂuctuations of the RABs, so their b oxes are simple rectangles. But in our setting, when a ( λ, χ ) -RAB is absorb ed or reﬂected by the barrier, it may ﬂuctuate as muc h as λ , whic h ma y b e v ery wildly , so w e need more complex boxes to contain it, and this mak es the gestion of the b oxes more complicated. F or any i ∈ { 1 , 2 } , n ∈ N ∗ , k ∈ N , y ∈ [ x − 5 − n ( k + 1) , x − 5 − n k ) , we deﬁne the “lo wer frontier of the b ox” a ( i, − ) n,k ( y ) =  Λ ∗ ( x,h i ) ( x − 5 − n k ) − 2 − n if λ ( z ) < Λ ∗ ( x,h i ) ( x − 5 − n k ) − 2 − n for all z ∈ ( y , x − 5 − n k ] , λ ( y ) − 2 − n otherwise. The second case w as introduced b ecause if Λ ∗ ( x,h i ) is to o close to the barrier, it may b e absorb ed b y it, hence the b ottom of the b o x needs to b e at least as low as the barrier. If y = x − 5 − n k , the deﬁnition is the same except that the condition “ λ ( z ) < Λ ∗ ( x,h i ) ( x − 5 − n k ) − 2 − n for all z ∈ ( y , x − 5 − n k ] ” is replaced by “ λ ( x − 5 − n k ) < Λ ∗ ( x,h i ) ( x − 5 − n k ) − 2 − n ”. F or y ∈ [ x − 5 − n ( k + 1) , x − 5 − n k ] , we also deﬁne the “upp er frontier of the b ox” a ( i, +) n,k ( y ) = max Λ ∗ ( x,h i ) ( x − 5 − n k ) + 2 − n , sup y ≤ z ≤ x − 5 − n k λ ( z ) + 2 − n ! . The second term in the maximum is there in case the barrier has a large jump and “pushes Λ ∗ ( x,h i ) up w ards”, to k eep the pro cess in the box. Our deﬁnition of the b ox A ( i ) n,k is { ( y , h ) ∈ R 2 | y ∈ ( x − 5 − n ( k + 1) , x − 5 − n k ] , a ( i, − ) n,k ( y ) < h < a ( i, +) n,k ( y ) } . F or i ∈ { 1 , 2 } , n ∈ N ∗ , w e also denote u ( i ) n = inf { y ≥ − x | ∃ k ∈ N , − y ∈ ( x − 5 − n ( k + 1) , x − 5 − n k ] , ( − y, Λ ∗ ( x,h i ) ( − y )) ∈ A ( i ) n,k } the ﬁrst moment a pro cess go es ab o v e the top of a b o x or b elow its b ottom, and v n = min {− x + 5 − n k | k ∈ N , A (1) n,k ∩ A (2) n,k  = ∅} the ﬁrst moment the b o xes are not disjoint anymore. Finally , w e set Y = inf { y ≥ − x | Λ ∗ ( x,h 1 ) ( − y ) = Λ ∗ ( x,h 2 ) ( − y ) } the meeting time of the pro cesses. In order to pro v e the bac kw ard lines are independent until they meet, w e replace Equations (9.31) and (9.32) in the proof of [32] (Pages 433 and 434) b y the following Lemmas 41 and 42. Lemma 41. F or i ∈ { 1 , 2 } , almost sur ely u ( i ) n tends to + ∞ when n tends to + ∞ . Lemma 42. Almost sur ely, v n tends to Y when n tends to + ∞ and v n ≤ Y when n is lar ge enough. Pr o of of L emma 41. Let i ∈ { 1 , 2 } . T o pro v e this lemma, w e need to sho w the ﬂuctuations of Λ ∗ ( x,h i ) are small enough that it sta ys in its b oxes, whic h we will control with the help of estimates on the ﬂuctuations of its driving TRUE SELF-REPELLING MOTION ABOVE A GENERAL BARRIER 21 Bro wnian motion. Let us introduce some notation. W e already kno w Λ ∗ ( x,h i ) ( − . ) is a ( λ ( − . ) , − χ ) -RAB starting from ( x, h ) ; w e can assume it is driven b y a Brownian motion ( W y ) y ≥− x . F or n ∈ N ∗ , k ∈ N , we deﬁne A i,n,k = {∀ y ∈ [ − x + 5 − n k , − x + 5 − n ( k + 1)] , | W y − W − x +5 − n k | ≤ 2 − n − 2 } . Let us pro v e that for K ∈ N ∗ , T 0 ≤ k ≤ 5 n K − 1 A i,n,k implies u ( i ) n ≥ − x + K . W e assume T 0 ≤ k ≤ 5 n K − 1 A i,n,k . F or all 0 ≤ k ≤ 5 n K − 1 , we alwa ys hav e ( x − 5 − n k , Λ ∗ ( x,h i ) ( x − 5 − n k )) ∈ A ( i ) n,k . F urthermore, for y ∈ ( − x + 5 − n k , − x + 5 − n ( k + 1)) , there are t wo p ossibilities. Either Λ ∗ ( x,h i ) ( − . ) do es not meet λ ( − . ) in [ − x + 5 − n k , y ) , whic h implies | Λ ∗ ( x,h i ) ( − y ) − Λ ∗ ( x,h i ) ( x − 5 − n k ) | = | W − y − W − x +5 − n k | ≤ 2 − n − 2 , thus in particular ( − y , Λ ∗ ( x,h i ) ( − y )) ∈ A ( i ) n,k . Or Λ ∗ ( x,h i ) ( − . ) has met λ ( − . ) in [ − x + 5 − n k , y ) , and we denote y ′ = inf { z ∈ [ − x + 5 − n k , y ) | Λ ∗ ( x,h i ) ( − z ) = λ ( − z ) } the meeting time. By what w e ha ve just shown, | Λ ∗ ( x,h i ) ( − y ′ ) − Λ ∗ ( x,h i ) ( x − 5 − n k ) | ≤ 2 − n − 2 , hence | λ ( − y ′ ) − Λ ∗ ( x,h i ) ( x − 5 − n k ) | ≤ 2 − n − 2 , which yields a ( i, − ) n,k ( − y ) = λ ( − y ) − 2 − n , therefore Λ ∗ ( x,h i ) ( − y ) > a ( i, − ) n,k ( − y ) . Moreo ver, Λ ∗ ( x,h i ) ( − y ) ≤ W y + sup y ′ ≤ z ≤ y ( λ ( − z ) − W z ) = sup y ′ ≤ z ≤ y ( λ ( − z ) + ( W y − W − x +5 − n k ) − ( W z − W − x +5 − n k )) ≤ sup − y ≤ z ≤ x − 5 − n k λ ( z ) + 2 − n − 1 < a ( i, +) n,k ( − y ) . W e deduce ( − y , Λ ∗ ( x,h i ) ( − y )) ∈ A ( i ) n,k . Consequently , T 0 ≤ k ≤ 5 n K − 1 A i,n,k implies u ( i ) n ≥ − x + K . F rom this w e deduce P n ∈ N ∗ P λ,χ ( u ( i ) n < − x + K ) ≤ P n ∈ N ∗ P λ,χ ( S 0 ≤ k ≤ 5 n K − 1 A c i,n,k ) . If for all K ∈ N ∗ the latter sum is ﬁnite, then the Borel-Cantelli lemma yields that for all K ∈ N ∗ , almost surely when n is large enough, u ( i ) n ≥ − x + K , which implies Lemma 41. So we only ha v e to pro v e that for all K ∈ N ∗ , w e ha v e P n ∈ N ∗ P λ,χ ( S 0 ≤ k ≤ 5 n K − 1 A c i,n,k ) < + ∞ . This comes from the fact that for n ∈ N ∗ , k ∈ N , the estimate on the Bro w- nian motion in Lemma 24 yields P λ,χ ( A c i,n,k ) ≤ 2 √ 2 v 5 − n 2 − n − 2 √ π exp( − 2 − 2 n − 4 2 v 5 − n ) , hence P n ∈ N ∗ P λ,χ ( S 0 ≤ k ≤ 5 n K − 1 A c i,n,k ) ≤ P n ∈ N ∗ 8 K 5 n/ 2 2 n q 2 v π exp( − 1 2 5 v ( 5 4 ) n ) < + ∞ , whic h ends the pro of of Lemma 41. □ Pr o of of L emma 42. The idea of the pro of is that before the pro cesses meet, they are separated b y at least some small distance, while the b oxes sta y very close to their resp ective pro cesses when n is large, so they will be disjoin t for large n . W e ﬁrst pro v e that almost surely , when n is large enough, v n ≤ Y . If Y = + ∞ , it holds true. If Y < + ∞ , we notice that if the t w o processes are still in their b oxes at Y , then these boxes are not disjoint. Moreov er, by Lemma 41, almost surely u (1) n and u (2) n tend to + ∞ when n tends to + ∞ , so when n is large enough, u (1) n > Y and u (2) n > Y , hence if k is suc h that Y ∈ [ − x + 5 − n k , − x + 5 − n ( k + 1)) , then ( − Y , Λ ∗ ( x,h 1 ) ( − Y )) ∈ A (1) n,k and ( − Y , Λ ∗ ( x,h 2 ) ( − Y )) ∈ A (2) n,k , hence A (1) n,k ∩ A (2) n,k  = ∅ , thus v n ≤ − x + 5 − n k ≤ Y . W e no w pro ve the almost sure conv ergence of v n to Y . W e assume Y < + ∞ ; let ℓ ∈ N ∗ , it is enough to pro ve that v n ≥ Y − 1 ℓ when n is large enough (if Y = + ∞ , we can use a similar argument to pro ve that v n ≥ ℓ when n is large enough). W e will ﬁrst con trol the ﬂuctuations of Λ ∗ ( x,h 1 ) ( − . ) , Λ ∗ ( x,h 2 ) ( − . ) and λ . One can pro v e as in Lemma 9.1(iv) of [32] that Λ ∗ ( x,h 1 ) ( − . ) , Λ ∗ ( x,h 2 ) ( − . ) are almost surely contin uous, whic h yields the existence of ε > 0 (random) so that for all y ∈ [ − x, Y − 1 2 ℓ ] , we ha v e | Λ ∗ ( x,h 1 ) ( − y ) − Λ ∗ ( x,h 2 ) ( − y ) | ≥ ε . Moreov er, λ ( − . ) is uniformly con tin uous on [ − x, Y − 1 2 ℓ ] , so there exists n 0 ∈ N so that for y , z ∈ [ − x, Y − 1 2 ℓ ] , if | y − z | ≤ 5 − n 0 , then | λ ( − y ) − λ ( − z ) | ≤ ε/ 6 . W e no w choose n ≥ n 0 , and k ∈ N so that − x + 5 − n ( k + 1) ≤ Y − 1 2 ℓ . W e are going to prov e that A (1) n,k ∩ A (2) n,k = ∅ if n is large enough (uniformly in k ). In order to pro v e A (1) n,k ∩ A (2) n,k = ∅ , we are going to prov e the upp er and low er frontiers of these b o xes sta y close to their resp ectiv e pro cesses. W e set i ∈ { 1 , 2 } , y ∈ [ x − 5 − n ( k + 1) , x − 5 − n k ] , and we study | a ( i, ± ) n,k ( y ) − Λ ∗ ( x,h i ) ( x − 5 − n k ) | . W e 22 LAURE MARÊCHÉ b egin with a ( i, +) n,k ( y ) = max(Λ ∗ ( x,h i ) ( x − 5 − n k ) + 2 − n , sup y ≤ z ≤ x − 5 − n k λ ( z ) + 2 − n ) . If a ( i, +) n,k ( y ) = Λ ∗ ( x,h i ) ( x − 5 − n k ) + 2 − n , w e ha v e | a ( i, +) n,k ( y ) − Λ ∗ ( x,h i ) ( x − 5 − n k ) | ≤ 2 − n . Otherwise, we hav e λ ( x − 5 − n k ) ≤ Λ ∗ ( x,h i ) ( x − 5 − n k ) ≤ sup y ≤ z ≤ x − 5 − n k λ ( z ) , with | sup y ≤ z ≤ x − 5 − n k λ ( z ) − λ ( x − 5 − n k ) | ≤ ε/ 6 , hence | sup y ≤ z ≤ x − 5 − n k λ ( z ) − Λ ∗ ( x,h i ) ( x − 5 − n k ) | ≤ ε/ 6 , th us | a ( i, +) n,k ( y ) − Λ ∗ ( x,h i ) ( x − 5 − n k ) | ≤ 2 − n + ε/ 6 . Therefore, in all cases | a ( i, +) n,k ( y ) − Λ ∗ ( x,h i ) ( x − 5 − n k ) | ≤ 2 − n + ε/ 6 . W e now consider a ( i, − ) n,k ( y ) . The argument is written for y  = x − 5 − n k , but the case y = x − 5 − n k is similar and easier. If λ ( z ) < Λ ∗ ( x,h i ) ( x − 5 − n k ) − 2 − n for all z ∈ ( y , x − 5 − n k ] , w e hav e | a ( i, − ) n,k ( y ) − Λ ∗ ( x,h i ) ( x − 5 − n k ) | = 2 − n . Otherwise, there exists z ∈ ( y , x − 5 − n k ] so that λ ( z ) ≥ Λ ∗ ( x,h i ) ( x − 5 − n k ) − 2 − n . W e also ha v e Λ ∗ ( x,h i ) ( x − 5 − n k ) − 2 − n ≥ λ ( x − 5 − n k ) − 2 − n , and | λ ( z ) − λ ( x − 5 − n k ) | ≤ ε/ 6 , hence | λ ( z ) − Λ ∗ ( x,h i ) ( x − 5 − n k ) | ≤ ε/ 6 +2 − n . F urthermore, we can write | a ( i, − ) n,k ( y ) − Λ ∗ ( x,h i ) ( x − 5 − n k ) | = | λ ( y ) − 2 − n − Λ ∗ ( x,h i ) ( x − 5 − n k ) | ≤ | λ ( y ) − λ ( z ) | + | λ ( z ) − Λ ∗ ( x,h i ) ( x − 5 − n k ) | + 2 − n ≤ ε/ 6 + ε/ 6 + 2 − n + 2 − n = ε/ 3 + 2 − n +1 . Thus in all cases | a ( i, − ) n,k ( y ) − Λ ∗ ( x,h i ) ( x − 5 − n k ) | ≤ ε/ 3 + 2 − n +1 . W e deduce that for any ( y , h ) ∈ A ( i ) n,k w e ha v e | h − Λ ∗ ( x,h i ) ( x − 5 − n k ) | ≤ ε/ 3 + 2 − n +1 . In addition, b y assumption | Λ ∗ ( x,h 1 ) ( x − 5 − n k ) − Λ ∗ ( x,h 2 ) ( x − 5 − n k ) | ≥ ε . Consequen tly , if n is large enough to ha v e 2 − n +2 < ε/ 3 , we obtain A (1) n,k ∩ A (2) n,k = ∅ . W e conclude that if n is large enough, for all k ∈ N so that − x + 5 − n ( k + 1) ≤ Y − 1 2 ℓ , w e ha ve v n ≥ − x + 5 − n ( k + 1) , th us if n is large enough, v n ≥ Y − 1 ℓ , which ends the proof of Lemma 42. □ 5.4. Coalescence of the bac kw ard lines. In this section, we prov e the backw ard lines coalesce when they meet. The idea of [32] to deal with this w as to notice that if t w o bac kward lines do not coalesce when they meet, there is a forw ard line Λ ( ˜ x n , ˜ h n ) b et ween them, hence the meeting point is on Λ ( ˜ x n , ˜ h n ) . Moreov er, there will be at least three separate forw ard lines coming out from the immediate vicinit y of the meeting p oin t: one b elow the lo w est bac kw ard line of the tw o, one abov e the topmost backw ard line, and one betw een the t w o lines. Therefore one only has to pro ve there cannot b e three suc h forw ard lines close to a Λ ( ˜ x n , ˜ h n ) . Ho w ever, there are tw o main problems in adapting the approach of [32] to our setting. The ﬁrst is that it requires a b ound on the probabilit y three forward lines with close starting p oin ts are separate. The estimate of [32] relied on the fact their barrier w as the abscissa axis, so we need to replace it. This is done in Lemma 44; how ev er, in the part of the space where the Bro wnian motions are absorb ed, the b ound only works a w a y from λ , which forces us to handle the pro of more carefully than in [32]. The second main problem is that the barrier ma y hav e very wild v ariations, hence Λ ( ˜ x n , ˜ h n ) ma y also v ary wildly , and if it do es, “close to Λ ( ˜ x n , ˜ h n ) ” may be in a zone to o wide to obtain go o d b ounds. In order to solve this problem, w e had to separate the cases depending on the behavior of λ . W e begin by form ulating the exact statement w e need to prov e. F or an y ( x, h ) ∈ R 2 λ , η > 0 , we denote O η ( x, h ) = lim ε → 0 card( { Λ ( x ′ ,h ′ ) ( x + η ) | ( x ′ , h ′ ) ∈ ( x, x + ε ) × ( h − ε, h + ε ) } ) the “n um b er of forward lines coming from the immediate vicinity of ( x, h ) which are still separate at x + η ”. As in [32], in order to pro v e bac kw ard lines coalesce when they meet, it is enough to pro v e the follo wing Proposition 43 (indeed, it allows to prov e an equiv alen t of Equation (9.38) of [32], whic h yields the result as in the pro of of their Equation (9.39)). Prop osition 43. F or any n ∈ N , η > 0 , M ∈ N ∗ ﬁxe d, almost sur ely for al l x ∈ [ ˜ x n , χ + M ) so that Λ ( ˜ x n , ˜ h n ) ( x ) > λ ( x ) we have O η ( x, Λ ( ˜ x n , ˜ h n ) ( x )) ≤ 2 . TRUE SELF-REPELLING MOTION ABOVE A GENERAL BARRIER 23 Before proving Prop osition 43, we ﬁrst state and prov e a b ound on the probability there exist three Λ ( x,h ) with their h close to eac h other whic h remain separate up to x + ε . This b ound is in the spirit of Lemma 9.4 of [32], but w orks only aw a y from λ in the part of the space with absorption. F or any ( x, h ) ∈ R 2 , ε > 0 , δ > 0 , we denote D ( x, h, h + δ , ε ) = {∃ h 1 , h 2 , h 3 ∈ [ h, h + δ ] , Λ ( x,h 1 ) ( x + ε ) < Λ ( x,h 2 ) ( x + ε ) < Λ ( x,h 3 ) ( x + ε ) } . If h ≤ λ ( x ) , then Λ ( x,h ′ ) is not deﬁned for h ′ ≤ λ ( x ) , and h 1 , h 2 , h 3 ∈ [ h, h + δ ] must b e understo o d as h 1 , h 2 , h 3 ∈ ( λ ( x ) , h + δ ] . Lemma 44. Ther e exists a c onstant C so that for any ( x, h ) ∈ R 2 λ , ε > 0 , δ > 0 , we have the fol lowing: • If λ ( x ) < h , P λ,χ ( {∀ y ∈ [ x, x + ε ] , Λ ( x,h ) ( y ) > λ ( y ) } ∩ D ( x, h, h + δ , ε )) ≤ C ( δ √ ε ) 3 . • If x + ε ≤ χ then P λ,χ ( D ( x, h, h + δ , ε )) ≤ C ( δ √ ε ) 3 . Pr o of. F or the ﬁrst p oin t, the idea is that if the Λ ( x,h i ) do not meet λ , they b ehav e as Bro wnian motions hence we can use Bro wnian motion estimates. F or the second point, the idea is that being reﬂected on λ will push the lo w est Λ ( x,h i ) to w ards the top, hence will only mak e it more likely to encounter the other Λ ( x,h i ) . W e will pro v e there exists a constant C ′ so that for any ( x, h ) ∈ R 2 λ , ε > 0 , δ > 0 , we hav e the following: (i) P λ,χ ( ∀ y ∈ [ x, x + ε ] , Λ ( x,h ) ( y ) > λ ( y ) , Λ ( x,h ) ( x + ε ) < Λ ( x,h + δ ) ( x + ε ) < Λ ( x,h +2 δ ) ( x + ε )) ≤ C ′ ( δ √ ε ) 3 . (ii) If x + ε ≤ χ then P λ,χ (Λ ( x,h ) ( x + ε ) < Λ ( x,h + δ ) ( x + ε ) < Λ ( x,h +2 δ ) ( x + ε )) ≤ C ′ ( δ √ ε ) 3 . Indeed, if w e ha v e the ab ov e, then Lemma 44 can be obtained for ( x, h ) ∈ R 2 λ through the pro of of Lemma 9.4 of [32], by substituting the abov e for Lemma A.1 of [32]. Moreo v er, if x + ε ≤ χ and h = λ ( x ) , we obtain Lemma 44 by writing D ( x, h, h + δ, ε ) as the increasing union of the D ( x, h + 1 n , h + δ, ε ) , n ∈ N ∗ . W e thus seek to pro ve (i) and (ii). T o do this, we need some notation. Let ( x, h ) ∈ R 2 λ , ε > 0 , δ > 0 . Let ( W 1 ( y )) y ≥ x , ( W 2 ( y )) y ≥ x , ( W 3 ( y )) y ≥ x b e indep endent Brownian motions so that W 1 ( x ) = h , W 2 ( x ) = h + δ , W 3 ( x ) = h + 2 δ . Let ( R 1 ( y )) y ≥ x , ( R 2 ( y )) y ≥ x , ( R 3 ( y )) y ≥ x b e the ( λ, χ ) -RABs starting from ( x, h ) , ( x, h + δ ) , ( x, h + 2 δ ) and driv en by ( W 1 ( y )) y ≥ x , ( W 2 ( y )) y ≥ x , ( W 3 ( y )) y ≥ x . Let Y = inf { y ≥ x | ∃ i, j ∈ { 1 , 2 , 3 } , R i ( y ) = R j ( y ) } the ﬁrst time at which t w o of the R i meet. It is enough to pro v e that P λ,χ ( ∀ y ∈ [ x, x + ε ] , R 1 ( y ) > λ ( y ) , Y > x + ε ) < C ′ ( δ √ ε ) 3 , and that if x + ε ≤ χ then P λ,χ ( Y > x + ε ) < C ′ ( δ √ ε ) 3 . Let Y ′ = inf { y ≥ x | ∃ i, j ∈ { 1 , 2 , 3 } , W i ( y ) = W j ( y ) } is the ﬁrst time at whic h tw o of the W i meet. If for all y ∈ [ x, x + ε ] we hav e R 1 ( y ) > λ ( y ) and Y > x + ε , then none of the R i , i ∈ { 1 , 2 , 3 } meets λ in [ x, x + ε ] , hence for all i ∈ { 1 , 2 , 3 } , y ∈ [ x, x + ε ] , w e ha ve R i ( y ) = W i ( y ) , whic h implies Y ′ ≥ x + ε . Consequently P λ,χ ( ∀ y ∈ [ x, x + ε ] , R 1 ( y ) > λ ( y ) , Y > x + ε ) ≤ P λ,χ ( Y ′ ≥ x + ε ) ≤ ˜ C ( δ √ ε ) 3 b y Lemma 26. W e no w consider the case x + ε ≤ χ . If there exists y ∈ [ x, x + ε ] so that R 2 ( y ) = λ ( y ) , then R 2 ( y ) ≤ R 1 ( y ) , hence Y ≤ y ≤ x + ε . Therefore if Y > x + ε , for all y ∈ [ x, x + ε ] w e ha v e R 2 ( y ) > λ ( y ) , hence R 2 ( y ) = W 2 ( y ) , and w e get R 3 ( y ) = W 3 ( y ) b y the same argumen t. In addition, for an y y ∈ [ x, x + ε ] w e ha v e W 1 ( y ) ≤ R 1 ( y ) . W e deduce that if Y > x + ε then Y ′ ≥ x + ε . This implies P λ,χ ( Y > x + ε ) ≤ ˜ C ( δ √ ε ) 3 b y Lemma 26, which ends the pro of of (i) and (ii), hence of Lemma 44. □ W e can no w pro ve Prop osition 43. Pr o of of Pr op osition 43. Let n ∈ N , η > 0 , M < + ∞ . W e will consider the case ˜ x n < χ as the case ˜ x n ≥ χ is similar and simpler. Let m ∈ N large, with 2 − m/ 3 < η / 2 . As in [32] (see their Equations (9.48) and (9.49)), w e will cut the tra jectory of Λ ( ˜ x n , ˜ h n ) in to interv als [ x j − 1 , x j ) of length muc h smaller than η , and we will con- struct “bad ev en ts” with small probability so that if there exists x ∈ [ x j − 1 , x j ) so that O η ( x, Λ ( ˜ x n , ˜ h n ) ( x )) > 2 , 24 LAURE MARÊCHÉ a bad ev en t o ccurs. In [32], the bad ev ent for x ∈ [ x j − 1 , x j ) w as of the form “ Λ ( x j − 1 , Λ ( ˜ x n , ˜ h n ) ( x j − 1 )+2 − m ) ( x j ) ≥ Λ ( ˜ x n , ˜ h n ) ( x j − 1 ) + 2 · 2 − m , or Λ ( x j − 1 , Λ ( ˜ x n , ˜ h n ) ( x j − 1 ) − 2 − m ) ( x j ) ≤ Λ ( ˜ x n , ˜ h n ) ( x j − 1 ) − 2 · 2 − m , or D ( x j , Λ ( ˜ x n , ˜ h n ) ( x j − 1 ) − 2 · 2 − m , Λ ( ˜ x n , ˜ h n ) ( x j − 1 ) + 2 · 2 − m , η / 2) ”. Indeed, if there exists x ∈ [ x j − 1 , x j ) so that O η ( x, Λ ( ˜ x n , ˜ h n ) ( x )) > 2 , there are three Λ ( x ′ ,h ′ ) with starting p oin ts close to ( x, Λ ( ˜ x n , ˜ h n ) ( x )) that do not coalesce b efore x + η . Then they start b et we en Λ ( x j − 1 , Λ ( ˜ x n , ˜ h n ) ( x j − 1 ) − 2 − m ) and Λ ( x j − 1 , Λ ( ˜ x n , ˜ h n ) ( x j − 1 )+2 − m ) , so their v alues at x j are b etw een Λ ( x j − 1 , Λ ( ˜ x n , ˜ h n ) ( x j − 1 ) − 2 − m ) ( x j ) and Λ ( x j − 1 , Λ ( ˜ x n , ˜ h n ) ( x j − 1 )+2 − m ) ( x j ) , whic h implies the bad even t (since in their case, at the right of x j , almost surely Λ ( x ′ ,h ′ ) = Λ ( x j , Λ ( x ′ ,h ′ ) ( x j )) ). Our bad even ts are notably more complex. Firstly , the estimate we ha v e on the probabilit y of D ( x j , Λ ( ˜ x n , ˜ h n ) ( x j − 1 ) − 2 · 2 − m , Λ ( ˜ x n , ˜ h n ) ( x j − 1 ) + 2 · 2 − m , η / 2) (Lemma 44) dep ends on whether w e are in the part of the space where Brownian motions are reﬂected or absorbed, so we ha ve to distinguish these tw o cases. In the part of the space with absorption, the estimate of Lemma 44 is v alid only aw a y from λ , hence we w ork in the part of the tra jectory of Λ ( ˜ x n , ˜ h n ) whic h is at distance 2 − k from λ , then mak e k tend to + ∞ . In the part of the space with reﬂection, if Λ ( ˜ x n , ˜ h n ) ( x j − 1 ) is to o close to λ ( x j − 1 ) , then Λ ( x j , Λ ( ˜ x n , ˜ h n ) ( x j − 1 ) − 2 − m ) ( x j ) may not b e deﬁned. [32] does not seem to notice the fact, but in their case all the Λ ( x ′ ,h ′ ) are ab ov e the abscissa axis, whic h allows to control their v alue at x j from b elow. Since our λ ma y ha v e a sharp decrease in [ x j − 1 , x j ) , the control it w ould giv e us is not suﬃcient. W e will th us separate into three cases: b eing aw a y from λ , in which case we w ork as in [32] (Page 437), b eing close to λ when λ has no sharp decrease, in which case saying the Λ ( x ′ ,h ′ ) are ab ov e λ at x j is suﬃcient, and b eing close to λ when λ has a sharp decrease. In this case, we consider (roughly) the Λ ( ˜ x, ˜ h ) starting from the p oint at whic h λ starts to decrease too muc h; then the Λ ( x ′ ,h ′ ) will b e ab ov e Λ ( ˜ x, ˜ h ) . Before deﬁning our bad even ts, we deﬁne the follo wing “go o d even ts”, whic h will o ccur almost surely and help us pro v e the bad ev en ts o ccur. G r,m = \ n ′ ∈ N \ x ∈ ˜ x n +5 − m N , ˜ x n ′ ≤ x<χ { if Λ ( ˜ x n ′ , ˜ h n ′ ) ( x ) > λ ( x ) then ∀ y ≥ x, Λ ( ˜ x n ′ , ˜ h n ′ ) ( y ) = Λ ( x, Λ ( ˜ x n ′ , ˜ h n ′ ) ( x )) ( y ) }∩ { if Λ ( ˜ x n ′ , ˜ h n ′ ) ( x ) = λ ( x ) then ∀ ε > 0 , ∃ h > λ ( x ) , ∀ y ≥ x + ε, Λ ( ˜ x n ′ , ˜ h n ′ ) ( y ) = Λ ( x,h ) ( y ) } , G a,m = \ n ′ ∈ N \ x ∈ χ +5 − m N , ˜ x n ′ ≤ x { if Λ ( ˜ x n ′ , ˜ h n ′ ) ( x ) > λ ( x ) then ∀ y ≥ x, Λ ( ˜ x n ′ , ˜ h n ′ ) ( y ) = Λ ( x, Λ ( ˜ x n ′ , ˜ h n ′ ) ( x )) ( y ) } . Lemma 45. G r,m and G a,m o c cur almost sur ely. Pr o of. Let n ′ ∈ N , x ∈ ( ˜ x n + 5 − m N ) ∪ ( χ + 5 − m N ) with ˜ x n ′ ≤ x . The fact that almost surely , if Λ ( ˜ x n ′ , ˜ h n ′ ) ( x ) > λ ( x ) then for all y ≥ x , Λ ( ˜ x n ′ , ˜ h n ′ ) ( y ) = Λ ( x, Λ ( ˜ x n ′ , ˜ h n ′ ) ( x )) ( y ) can b e prov en as in Lemma 8.3 of [32], by “squeezing” Λ ( x, Λ ( ˜ x n ′ , ˜ h n ′ ) ( x )) b et ween t wo sequences Λ ( x,h ( n )) , Λ ( x,h ′ ( n )) with h ( n ) < Λ ( ˜ x n ′ , ˜ h n ′ ) ( x ) < h ′ ( n ) so that Λ ( x,h ( n )) and Λ ( x,h ′ ( n )) coalesce quic kly after x ; to estimate the probabilit y Λ ( x,h ( n )) and Λ ( x,h ′ ( n )) coalesce, w e can use an argument similar to that of Lemma 31. If w e also hav e x < χ , { if Λ ( ˜ x n ′ , ˜ h n ′ ) ( x ) = λ ( x ) then ∀ ε > 0 , ∃ h > λ ( x ) , ∀ y ≥ x + ε, Λ ( ˜ x n ′ , ˜ h n ′ ) ( y ) = Λ ( x,h ) ( y ) } happ ens almost surely thanks to Lemma 38 (using a sequence of ε tending to 0), whic h ends the pro of of the lemma. □ As we already men tioned, the pro of of Prop osition 43 dep ends on whether w e are in the part of the space where the Brown ian motions are reﬂected, at the left of χ , or in the part of the space where they are absorbed, at the right of χ . W e b egin with the latter, as the pro of there is simpler. TRUE SELF-REPELLING MOTION ABOVE A GENERAL BARRIER 25 Part of the sp ac e with absorption. W e in tro duce bad ev en ts so that if there exists x ∈ [ χ, χ + M ) so that Λ ( ˜ x n , ˜ h n ) ( x ) > λ ( x ) and O η ( x, Λ ( ˜ x n , ˜ h n ) ( x )) > 2 , then one of these bad ev en ts o ccurs. Let k large and Y k = inf { x ≥ χ | min y ∈ [ x,x +2 − k ] | Λ ( ˜ x n , ˜ h n ) ( x ) − λ ( y ) | ≤ 2 − k } . The follo wing quan tities and even ts will dep end on m and sometimes k , but we do not reﬂect this in the notation to mak e it ligh ter. F or all j ∈ { 0 , ..., ⌊ 5 m M ⌋} , we denote x j = χ + 5 − m j , h j = Λ ( ˜ x n , ˜ h n ) ( x j ) . F or j ∈ { 1 , ..., ⌊ 5 m M ⌋} , the bad ev en t A j is deﬁned as follows. A j = { x j − 1 < Y k } ∩  D ( x j , h j − 1 − 2 − m +1 , h j − 1 + 2 − m +1 , 2 − m 3 ) ∪ { Λ ( x j − 1 ,h j − 1 +2 − m ) ( x j ) ≥ h j − 1 + 2 − m +1 } ∪ { Λ ( ˜ x n , ˜ h n ) ( x j ) = Λ ( x j − 1 ,h j − 1 +2 − m ) ( x j ) } ∪{ Λ ( x j − 1 ,h j − 1 − 2 − m ) ( x j ) ≤ h j − 1 − 2 − m +1 } ∪ { Λ ( ˜ x n , ˜ h n ) ( x j ) = Λ ( x j − 1 ,h j − 1 − 2 − m ) ( x j ) }  . The even ts { Λ ( ˜ x n , ˜ h n ) ( x j ) = Λ ( x j − 1 ,h j − 1 +2 − m ) ( x j ) } and { Λ ( ˜ x n , ˜ h n ) ( x j ) = Λ ( x j − 1 ,h j − 1 − 2 − m ) ( x j ) } are not present in [32], but they seem necessary , as if one of them happ ens, the v alue of Λ ( x ′ ,h ′ ) with x ′ at the right of the coalescence is not con trolled b y Λ ( x j − 1 ,h j − 1 − 2 − m ) ( x j ) . The ev en t { x j − 1 < Y k } is there to ensure we are not to o close to λ , where Lemma 44 fails. W e hav e the follo wing lemmas. Lemma 46. When m is lar ge enough, for j ∈ { 1 , ..., ⌊ 5 m M ⌋} , almost sur ely if x j − 1 < Y k and ther e exists x ∈ [ x j − 1 , x j ) so that O η ( x, Λ ( ˜ x n , ˜ h n ) ( x )) > 2 , then A j o c curs. Lemma 47. When m is lar ge enough, for any j ∈ { 1 , ..., ⌊ 5 m M ⌋} , P λ,χ ( A j ) ≤ 6 exp( − 5 m 2 2 m +5 v ) + exp( − 1 v 2 m 3 − 2 k − 5 ) + C 2 − 17 6 m +6 . Lemmas 46 and 47 are enough to pro ve that almost surely for all x ∈ [ χ, χ + M ) so that Λ ( ˜ x n , ˜ h n ) ( x ) > λ ( x ) , w e ha v e O η ( x, Λ ( ˜ x n , ˜ h n ) ( x )) ≤ 2 , whic h is one half of Proposition 43. Indeed, when m is large, Lemma 46 yields that almost surely , if there exists x ∈ [ χ, min( Y k , χ + M − 5 − m )) so that O η ( x, Λ ( ˜ x n , ˜ h n ) ( x )) > 2 , there exists j ∈ { 1 , ..., ⌊ 5 m M ⌋} so that A j o ccurs. Hence almost surely {∃ x ∈ [ χ, min( Y k , χ + M )) , O η ( x, Λ ( ˜ x n , ˜ h n ) ( x )) > 2 } ⊂ S m ′ T m ≥ m ′ S j ∈{ 1 ,..., ⌊ 5 m M ⌋} A j . Moreo ver, Lemma 47 implies that w e hav e P λ,χ ( S m ′ T m ≥ m ′ S j ∈{ 1 ,..., ⌊ 5 m M ⌋} A j ) = lim m ′ → + ∞ P λ,χ ( T m ≥ m ′ S j ∈{ 1 ,..., ⌊ 5 m M ⌋} A j ) ≤ lim m ′ → + ∞ (5 m ′ M (6 exp( − 5 m ′ 2 2 m ′ +5 v ) + exp( − 1 v 2 m ′ 3 − 2 k − 5 ) + C 2 − 17 6 m ′ +6 )) = 0 . Therefore almost surely for all x ∈ [ χ, min( Y k , χ + M )) , O η ( x, Λ ( ˜ x n , ˜ h n ) ( x )) ≤ 2 . By making k tend to + ∞ we obtain that almost surely for all x ∈ [ χ, χ + M ) so that Λ ( ˜ x n , ˜ h n ) ( x ) > λ ( x ) , w e ha v e O η ( x, Λ ( ˜ x n , ˜ h n ) ( x )) ≤ 2 . W e no w pro v e lemmas 46 and 47. Pr o of of L emma 46. This can be seen easily , so w e only giv e a sketc h. Let m m uc h larger than k , j ∈ { 1 , ..., ⌊ 5 m M ⌋} so that x j − 1 < Y k and there exists x ∈ [ x j − 1 , x j ) so that O η ( x, Λ ( ˜ x n , ˜ h n ) ( x )) > 2 . Then for ε small there exist three Λ ( x ′ ,h ′ ) with ( x ′ , h ′ ) ∈ ( x, x + ε ) × ( h − ε, h + ε ) that do not merge before x ′ + η . If Λ ( ˜ x n , ˜ h n ) ( x j )  = Λ ( x j − 1 ,h j − 1 +2 − m ) ( x j ) , then Λ ( ˜ x n , ˜ h n ) is strictly below Λ ( x j − 1 ,h j − 1 +2 − m ) around x , hence the Λ ( x ′ ,h ′ ) are b elow Λ ( x j − 1 ,h j − 1 +2 − m ) . There- fore, if Λ ( x j − 1 ,h j − 1 +2 − m ) ( x j ) < h j − 1 + 2 − m +1 , the Λ ( x ′ ,h ′ ) ( x j ) are smaller than h j − 1 + 2 − m +1 . In the same w a y , if Λ ( x j − 1 ,h j − 1 − 2 − m ) ( x j ) > h j − 1 − 2 − m +1 and Λ ( ˜ x n , ˜ h n ) ( x j )  = Λ ( x j − 1 ,h j − 1 − 2 − m ) ( x j ) , the Λ ( x ′ ,h ′ ) ( x j ) are bigger than h j − 1 − 2 − m +1 . Moreo v er, we hav e the third p oint of Theorem 7, b y Lemma 45 we hav e that almost surely G a,m o ccurs, and by assumption x j − 1 < Y k , therefore almost surely D ( x j , h j − 1 − 2 − m +1 , h j − 1 + 2 − m +1 , 2 − m 3 ) o ccurs. □ 26 LAURE MARÊCHÉ Pr o of of L emma 47. The estimate on the probabilit y of the bad even t A j will come from the fact that if it o ccurs, either a “ D ev en t” o ccurs, whic h can b e dealt with thanks to Lemma 44, or the forw ard lines ha v e large ﬂuctuations, whic h can b e treated using Lemma 27. W e ﬁrst introduce an additional notation. If G a,m o ccurs (whic h happ ens almost surely thanks to Lemma 45), w e can write A j ⊂ { min y ∈ [ x j − 1 ,x j − 1 +2 − k ] | h j − 1 − λ ( y ) | > 2 − k } ∩ A j ( h j − 1 ) , where for any h > λ ( x j − 1 ) so that min y ∈ [ x j − 1 ,x j − 1 +2 − k ] | h − λ ( y ) | > 2 − k , we denote A j ( h ) = D ( x j , h − 2 − m +1 , h + 2 − m +1 , 2 − m 3 ) ∪ { Λ ( x j − 1 ,h +2 − m ) ( x j ) ≥ h + 2 − m +1 } ∪ { Λ ( x j − 1 ,h ) ( x j ) = Λ ( x j − 1 ,h +2 − m ) ( x j ) } ∪ { Λ ( x j − 1 ,h − 2 − m ) ( x j ) ≤ h − 2 − m +1 } ∪ { Λ ( x j − 1 ,h ) ( x j ) = Λ ( x j − 1 ,h − 2 − m ) ( x j ) } . W e will prov e that for any h > λ ( x j − 1 ) so that min y ∈ [ x j − 1 ,x j − 1 +2 − k ] | h − λ ( y ) | > 2 − k , A j ( h ) is indep endent from h j − 1 . Indeed, if w e construct a sequence ( x n ′ , h n ′ ) , n ′ ∈ N so that ( x 0 , h 0 ) = ( ˜ x n , ˜ h n ) , for all n ′ ≥ 1 odd, x n ′ = x j − 1 , for all n ′ ≥ 1 ev en, x n ′ = x j , the set { h n ′ | n ′ ≥ 1 odd } is dense in [ λ ( x j − 1 ) , + ∞ ) and { h n ′ | n ′ ≥ 1 ev en } is dense in [ λ ( x j ) , + ∞ ) , then for an y N < + ∞ , the family (Λ ( x n ′ ,h n ′ ) ) 0 ≤ n ′ ≤ N is a ( λ, χ ) - FICRAB. Consequently (Λ ( ˜ x n , ˜ h n ) ( y )) ˜ x n ≤ y ≤ x j − 1 is indep enden t from the ((Λ ( x j − 1 ,h n ′ ) ) n ′ ≥ 1 o dd , (Λ ( x j ,h n ′ ) ) n ′ ≥ 1 even ) , th us from the ((Λ ( x j − 1 ,h ′ ) ) h ′ >λ ( x j − 1 ) , (Λ ( x j ,h ′ ) ) h ′ >λ ( x j ) ) thanks to the left-contin uit y of the pro cess. This allo ws to show A j ( h ) is independent from h j − 1 . Hence to pro ve Lemma 47, it is enough to prov e that for any h > λ ( x j − 1 ) so that min y ∈ [ x j − 1 ,x j − 1 +2 − k ] | h − λ ( y ) | > 2 − k , we hav e P λ,χ ( A j ( h )) ≤ 6 exp( − 5 m 2 2 m +5 v ) + exp( − 1 v 2 m 3 − 2 k − 5 ) + C 2 − 17 6 m +6 . W e no w study P λ,χ ( A j ( h )) for h > λ ( x j − 1 ) with min y ∈ [ x j − 1 ,x j − 1 +2 − k ] | h − λ ( y ) | > 2 − k . By Lemma 27, we hav e P λ,χ (Λ ( x j − 1 ,h +2 − m ) ( x j ) ≥ h + 2 − m +1 ) ≤ P λ,χ ( ∃ x ∈ [ x j − 1 , x j ] , | Λ ( x j − 1 ,h +2 − m ) ( x ) − Λ ( x j − 1 ,h +2 − m ) ( x j − 1 ) | ≥ 2 − m ) ≤ exp( − 5 m 2 2 m +1 v ) when m is large enough, and similarly P λ,χ (Λ ( x j − 1 ,h − 2 − m ) ( x j ) ≤ h − 2 − m +1 ) ≤ exp( − 5 m 2 2 m +1 v ) when m is large enough. F urthermore, Λ ( x j − 1 ,h ) ( x j ) < Λ ( x j − 1 ,h +2 − m ) ( x j ) if w e ha v e max x ∈ [ x j − 1 ,x j ] | Λ ( x j − 1 ,h ) ( x ) − Λ ( x j − 1 ,h ) ( x j − 1 ) | ≤ 2 − m − 2 and max x ∈ [ x j − 1 ,x j ] | Λ ( x j − 1 ,h +2 − m ) ( x ) − Λ ( x j − 1 ,h +2 − m ) ( x j − 1 ) | ≤ 2 − m − 2 . By Lemma 27, the probabilit y that one of these maxima is bigger than 2 − m − 2 is smaller than exp( − 5 m 2 2 m +5 v ) when m is large enough, therefore P λ,χ (Λ ( x j − 1 ,h ) ( x j ) = Λ ( x j − 1 ,h +2 − m ) ( x j )) ≤ 2 exp( − 5 m 2 2 m +5 v ) when m is large enough. The same arguments yield that P λ,χ (Λ ( x j − 1 ,h ) ( x j ) = Λ ( x j − 1 ,h − 2 − m ) ( x j )) ≤ 2 exp( − 5 m 2 2 m +5 v ) when m is large enough. Moreov er, when m is large enough, we obtain P λ,χ ( ∃ x ∈ [ x j , x j + 2 − m 3 ] , Λ ( x j ,h − 2 − m +1 ) ( x ) = λ ( x )) ≤ P λ,χ ( ∃ x ∈ [ x j , x j + 2 − m 3 ] , | Λ ( x j ,h − 2 − m +1 ) ( x ) − Λ ( x j ,h − 2 − m +1 ) ( x j ) | ≥ 2 − k − 2 ) ≤ exp( − 1 v 2 m 3 − 2 k − 5 ) b y Lemma 27. Finally , Lemma 44 yields P λ,χ ( {∀ x ∈ [ x j , x j + 2 − m 3 ] , Λ ( x j ,h − 2 − m +1 ) ( x ) > λ ( x ) } ∩ D ( x j , h − 2 − m +1 , h + 2 − m +1 , 2 − m 3 ) } ) ≤ C (2 − m +2 / p 2 − m 3 ) 3 = C 2 − 17 6 m +6 . This is enough to prov e Lemma 47. □ Part of the sp ac e with r eﬂe ction. W e will deﬁne bad ev en ts so that if there exists x ∈ [ ˜ x n , χ ) so that O η ( x, Λ ( ˜ x n , ˜ h n ) ( x )) > 2 , one of these bad ev en ts o ccurs. W e will reuse some notations already used in the part of the space with absorption, but with a diﬀerent meaning, as using diﬀerent notations would make them m uc h hea vier and harder to read. F or an y j ∈ { 0 , ..., ⌊ 5 m ( χ − 2 − m/ 3 − ˜ x n ) ⌋} , w e denote x j = ˜ x n + 5 − m j , h j = Λ ( ˜ x n , ˜ h n ) ( x j ) . F or j ∈ { 1 , ..., ⌊ 5 m ( χ − 2 − m/ 3 − ˜ x n ) ⌋} , w e denote m j = max x ∈ [ x j − 1 ,x j ] λ ( x ) and deﬁne M 1 j = { h j − 1 − 2 − m +1 > m j } . On this ev ent, we are “a w ay from λ ” and can use arguments similar to those used in the part of the space with absorption. W e no w denote M 2 j = ( M 1 j ) c ∩ { λ ( x j ) ≥ m j − 2 − m } . On this even t, we are close to λ , but λ do es not go do wn to o muc h after its maxim um, so λ can bound from b elow the forw ard lines starting close to Λ ( ˜ x n , ˜ h n ) . W e also deﬁne M 3 j = ( M 1 j ) c ∩ { λ ( x j ) < m j − 2 − m } , in which case the lo wer b ound m ust be diﬀerent. Let us denote x ′ j = sup { x ∈ [ x j − 1 , x j ] | λ ( x ) = m j } , x ′′ j = sup { x ≤ x j | λ ( x ) = m j − 2 − m } , and h ′ j = max( h j − 1 + 2 − m , m j + 2 − m +1 ) . On the ev ent M 3 j , the low er b ound we w ould like to use for the forw ard TRUE SELF-REPELLING MOTION ABOVE A GENERAL BARRIER 27 lines starting close to Λ ( ˜ x n , ˜ h n ) is the forward line starting from ( x ′′ j , λ ( x ′′ j )) , but it is not deﬁned, so instead w e use a Λ ( ˜ x n ′ , ˜ h n ′ ) so that Λ ( ˜ x n ′ , ˜ h n ′ ) ( x ′′ j ) = λ ( x ′′ j ) , if there is one, or inf ℓ ∈ N ∗ Λ ( x ′′ j ,λ ( x ′′ j )+1 /ℓ ) otherwise. Our bad even t A j is constructed as A j = A 0 j ∪ ( M 1 j ∩ A 1 j ) ∪ ( M 2 j ∩ A 2 j ) ∪ ( M 3 j ∩ A 3 j ) with A 0 j = { Λ ( x j − 1 ,h ′ j ) ( x j ) ≥ h ′ j + 2 − m } ∪ { Λ ( ˜ x n , ˜ h n ) ( x j ) = Λ ( x j − 1 ,h ′ j ) ( x j ) } , A 1 j = { Λ ( x j − 1 ,h j − 1 − 2 − m ) ( x j ) ≤ h j − 1 − 2 − m +1 }∪{ Λ ( ˜ x n , ˜ h n ) ( x j ) = Λ ( x j − 1 ,h j − 1 − 2 − m ) ( x j ) }∪D ( x j , h j − 1 − 2 − m +1 , h ′ j +2 − m , 2 − m 3 ) , A 2 j = D ( x j , λ ( x j ) , h ′ j + 2 − m , 2 − m 3 ) , A 3 j = { Λ ( ˜ x n , ˜ h n ) ( x ′′ j ) = λ ( x ′′ j ) } ∪ {∃ n ′ ∈ N , Λ ( ˜ x n ′ , ˜ h n ′ ) ( x ′′ j ) = λ ( x ′′ j ) , Λ ( ˜ x n ′ , ˜ h n ′ ) ( x j ) ≤ m j − 2 − m +1 }∪  inf ℓ ∈ N ∗ Λ ( x ′′ j ,λ ( x ′′ j )+1 /ℓ ) ( x j ) ≤ m j − 2 − m +1 or Λ ( ˜ x n , ˜ h n ) ( x j ) = inf ℓ ∈ N ∗ Λ ( x ′′ j ,λ ( x ′′ j )+1 /ℓ ) ( x j )  ∪ D ( x j , m j − 2 − m +1 , h ′ j + 2 − m , 2 − m 3 ) . A 0 j is inv olv ed with the upp er bound on forw ard lines starting near Λ ( ˜ x n , ˜ h n ) , the rest of the bad even t with the low er b ound. The last parameter in the even ts D is 2 − m 3 instead of η / 2 as in [32] b ecause the estimate on P ( D ( x, h, h + δ, ε )) giv en in Lemma 44 which holds for λ ( x ) = h is only v alid for x + ε ≤ χ , so we can work only with x j + ε ≤ χ . Cho osing ε = 2 − m 3 allo ws to make x j tend to χ when m tends to + ∞ . Our bad even ts will satisfy the following Lemmas 48 and 49. A pro of similar to the one used in the part of the space with absorption shows that if Lemmas 48 and 49 hold, almost surely for any x ∈ [ ˜ x n , χ ) w e ha v e O η ( x, Λ ( ˜ x n , ˜ h n ) ( x )) ≤ 2 , whic h is the second half of Prop osition 43. Therefore we only hav e to prov e these t wo lemmas to end the pro of of Prop osition 43. Lemma 48. A lmost sur ely, for any j ∈ { 1 , ..., ⌊ 5 m ( χ − 2 − m/ 3 − ˜ x n ) ⌋} , if ther e exists x ∈ [ x j − 1 , x j ) so that we have O η ( x, Λ ( ˜ x n , ˜ h n ) ( x )) > 2 , then A j o c curs. Lemma 49. When m is lar ge enough, for any j ∈ { 1 , ..., ⌊ 5 m ( χ − 2 − m/ 3 − ˜ x n ) ⌋} , we have P λ,χ ( A j ) ≤ 10 exp( − 5 m 2 2 m +5 v )+ C 2 10 2 − 5 2 m . Pr o of of L emma 48. This lemma can b e seen easily , so w e only provide a sk etc h. The existence of x ∈ [ x j − 1 , x j ) so that O η ( x, Λ ( ˜ x n , ˜ h n ) ( x )) > 2 w ould imply the existence of three Λ ( x ′ ,h ′ ) with ( x ′ , h ′ ) ∈ ( x, x + ε ) × ( h − ε, h + ε ) that do not merge b efore x + η . If Λ ( x j − 1 ,h ′ j ) has not merged with Λ ( ˜ x n , ˜ h n ) b efore x j , it will be strictly ab o v e Λ ( ˜ x n , ˜ h n ) , hence ab o ve these Λ ( x ′ ,h ′ ) , whic h are thus b elow Λ ( x j − 1 ,h ′ j ) . These Λ ( x ′ ,h ′ ) m ust also b e ab ov e the follo wing (again if there is no merging): Λ ( x j − 1 ,h j − 1 − 2 − m ) in the case M 1 j , λ in the case M 2 j , and inf ℓ ∈ N ∗ Λ ( x ′′ j ,λ ( x ′′ j )+1 /ℓ ) or Λ ( ˜ x n ′ , ˜ h n ′ ) in the case M 3 j . Given our b ounds on the v alues of these pro cesses, the v alues of the Λ ( x ′ ,h ′ ) at x j are in a “small interv al” whic h dep ends on the case. Moreo v er, w e ha v e the third p oint of Theorem 7, and that by Lemma 45 almost surely G r,m o ccurs, which yields three pro cesses Λ ( x j ,h 1 ) , Λ ( x j ,h 2 ) , Λ ( x j ,h 3 ) , with h 1 , h 2 , h 3 in the “small in terv al”, that do not merge b efore x ′ + η . □ Pr o of of L emma 49. As in the pro of of Lemma 47, w e will notice that if A j o ccurs, either a “ D ev en t” o ccurs (which can be dealt with thanks to Lemma 44), or the forward lines ﬂuctuate a lot. How ever, these ﬂuctuations are harder to control than in Lemma 47. Indeed, when the forward lines are a w a y from λ , Lemma 27 gives straigh tforward b ounds, but when the forw ard lines are close to λ we hav e to use several tric ks to link their ﬂuctuations to those of the Bro wnian motion driving them. Since M 3 j ∩ A 3 j in v olves an inﬁnite n um b er of forw ard lines, it requires a particularly delicate treatment. 28 LAURE MARÊCHÉ W e ﬁrst b ound P λ,χ ( A 0 j ) . F or an y h ≥ λ ( x j − 1 ) , w e denote h ′ = max( h + 2 − m , m j + 2 − m +1 ) and A 0 ( h ) = { Λ ( x j − 1 ,h ′ ) ( x j ) ≥ h ′ + 2 − m } ∪ { Λ ( x j − 1 ,h +2 − m − 2 ) ( x j ) = Λ ( x j − 1 ,h ′ ) ( x j ) } . W e ha v e A 0 j ⊂ A 1 ( h j − 1 ) . As in the pro of of Lemma 47, for an y h ≥ λ ( x j − 1 ) , w e hav e h j − 1 and A 0 ( h ) independent, hence it is enough to b ound P λ,χ ( A 0 ( h )) for an y such h . Lemma 27 yields P λ,χ (max y ∈ [ x j − 1 ,x j ] | Λ ( x j − 1 ,h ′ ) ( y ) − Λ ( x j − 1 ,h ′ ) ( x j − 1 ) | ≥ 2 − m ) ≤ exp( − 5 m 2 2 m +1 v ) when m is large enough (as h ′ is c hosen w ell ab ov e λ ), hence P λ,χ (Λ ( x j − 1 ,h ′ ) ( x j ) ≥ h ′ + 2 − m ) ≤ exp( − 5 m 2 2 m +1 v ) when m is large enough. Moreov er, if Λ ( x j − 1 ,h +2 − m − 2 ) ( x j ) = Λ ( x j − 1 ,h ′ ) ( x j ) , then max y ∈ [ x j − 1 ,x j ] | Λ ( x j − 1 ,h ′ ) ( y ) − Λ ( x j − 1 ,h ′ ) ( x j − 1 ) | ≥ 2 − m − 1 or max y ∈ [ x j − 1 ,x j ] Λ ( x j − 1 ,h +2 − m − 2 ) ( y ) ≥ h ′ − 2 − m − 1 . The probability of the ﬁrst ev ent is smaller than exp( − 5 m 2 2 m +3 v ) when m is large enough b y Lemma 27. W e now consider the second one. Let ( W y ) y ≥ x j the Brownian motion driving Λ ( x j − 1 ,h +2 − m − 2 ) . If max y ∈ [ x j − 1 ,x j ] Λ ( x j − 1 ,h +2 − m − 2 ) ( y ) ≥ h ′ − 2 − m − 1 , then max y ∈ [ x j − 1 ,x j ] | W y − W x j − 1 | ≥ 2 − m − 2 . Indeed, if Λ ( x j − 1 ,h +2 − m − 2 ) do es not meet λ b efore hitting the maximum it is ob vious, and if it do es, let z 1 the ﬁrst p oint at whic h the maxim um is reached and z 2 = sup { y ≤ z 1 | Λ ( x j − 1 ,h +2 − m − 2 ) ( y ) = λ ( y ) } , then W z 2 − W z 1 ≥ 2 − m . W e deduce P λ,χ (max y ∈ [ x j − 1 ,x j ] Λ ( x j − 1 ,h +2 − m − 2 ) ( y ) ≥ h ′ − 2 − m − 1 ) ≤ P λ,χ (max y ∈ [ x j − 1 ,x j ] | W y − W x j − 1 | ≥ 2 − m − 2 ) ≤ exp( − 5 m 2 2 m +5 v ) when m is large enough b y Lemma 24. Consequen tly , when m is large enough, P λ,χ ( A 0 ( h )) ≤ 3 exp( − 5 m 2 2 m +5 v ) , hence P λ,χ ( A 0 j ) ≤ 3 exp( − 5 m 2 2 m +5 v ) . W e no w b ound P λ,χ ( M 1 j ∩ A 1 j ) . Again, for any h ≥ λ ( x j − 1 ) , if h − 2 − m +1 > m j , w e deﬁne the ev en t A 1 ( h ) = { Λ ( x j − 1 ,h − 2 − m ) ( x j ) ≤ h − 2 − m +1 } ∪ { Λ ( x j − 1 ,h ) ( x j ) = Λ ( x j − 1 ,h − 2 − m ) ( x j ) } ∪ D ( x j , h − 2 − m +1 , h + 2 − m +1 , 2 − m 3 ) . If G r,m o ccurs (which happ ens almost surely by Lemma 45), we ha v e M 1 j ∩ A 1 j ⊂ { h j − 1 − 2 − m +1 > m j } ∩ A 1 ( h j − 1 ) . As in the pro of of Lemma 47, for any h with h − 2 − m +1 > m j , h j − 1 and A 1 ( h ) are independent, hence it is enough to b ound P λ,χ ( A 1 ( h )) for an y such h . W e b egin b y noticing the incremen ts of Λ ( x j − 1 ,h − 2 − m ) on [ x j − 1 , x j ] are abov e those of the Bro wnian motion driving it, hence if ( W y ) y ≥ x j − 1 is a Brownian motion, b y Lemma 24 we ha v e P λ,χ (Λ ( x j − 1 ,h − 2 − m ) ( x j ) ≤ h − 2 − m +1 ) ≤ P λ,χ ( W x j − W x j − 1 ≤ − 2 − m ) ≤ exp( − 5 m 2 2 m +1 v ) when m is large enough. W e no w study P λ,χ (Λ ( x j − 1 ,h ) ( x j ) = Λ ( x j − 1 ,h − 2 − m ) ( x j )) . If max y ∈ [ x j − 1 ,x j ] | Λ ( x j − 1 ,h ) ( y ) − Λ ( x j − 1 ,h ) ( x j − 1 ) | ≤ 2 − m − 2 and max y ∈ [ x j − 1 ,x j ] | Λ ( x j − 1 ,h − 2 − m ) ( y ) − Λ ( x j − 1 ,h − 2 − m ) ( x j − 1 ) | ≤ 2 − m − 2 , then Λ ( x j − 1 ,h ) ( x j )  = Λ ( x j − 1 ,h − 2 − m ) ( x j ) . Moreov er, by Lemma 27, the probability one of these maxima is bigger than 2 − m − 2 is smaller than exp( − 5 m 2 2 m +5 v ) when m is large enough, hence P λ,χ (Λ ( x j − 1 ,h ) ( x j ) = Λ ( x j − 1 ,h − 2 − m ) ( x j )) ≤ 2 exp( − 5 m 2 2 m +5 v ) . In addition, Lemma 44 yields P λ,χ ( D ( x j , h − 2 − m +1 , h + 2 − m +1 , 2 − m 3 )) ≤ C (2 − m +2 / p 2 − m 3 ) 3 = C 2 − 5 2 m +6 . W e thus obtain P λ,χ ( A 1 ( h )) ≤ 3 exp( − 5 m 2 2 m +5 v ) + C 2 − 5 2 m +6 , therefore P λ,χ ( M 1 j ∩ A 1 j ) ≤ 3 exp( − 5 m 2 2 m +5 v ) + C 2 − 5 2 m +6 . W e no w b ound P λ,χ ( M 2 j ∩ A 2 j ) . If M 2 j o ccurs we ha ve h ′ j + 2 − m ≤ m j + 2 − m +2 ≤ λ ( x j ) + 2 − m +2 + 2 − m , so P λ,χ ( M 2 j ∩ A 2 j ) ≤ P λ,χ ( D ( x j , λ ( x j ) , λ ( x j ) + 3 · 2 − m +1 , 2 − m 3 )) ≤ C (3 · 2 − m +1 / p 2 − m 3 ) 3 = C 6 3 2 − 5 2 m b y Lemma 44. W e ﬁnally bound P λ,χ ( M 3 j ∩ A 3 j ) . If M 3 j ∩ A 3 j o ccurs, then x ′′ j ∈ [ x ′ j , x j ] , for all y ∈ [ x ′′ j , x j ] we hav e λ ( y ) ≤ m j − 2 − m , and the following ev en t o ccurs: { min y ∈ [ x ′ j ,x j ] Λ ( ˜ x n , ˜ h n ) ( y ) < m j − 2 − m − 2 } ∪ {∃ n ′ ∈ N , Λ ( ˜ x n ′ , ˜ h n ′ ) ( x ′′ j ) = λ ( x ′′ j ) , Λ ( ˜ x n ′ , ˜ h n ′ ) ( x j ) ≤ m j − 2 − m +1 } ∪ { inf ℓ ∈ N ∗ Λ ( x ′′ j ,λ ( x ′′ j )+1 /ℓ ) ( x j ) ≤ m j − 2 − m +1 } ∪ { Λ ( x ′′ j ,λ ( x ′′ j )+2 − m − 2 ) ( x j ) > m j − 2 − m − 1 } ∪ D ( x j , m j − 2 − m +1 , m j + 2 − m +2 , 2 − m 3 ) , so it is enough to b ound the probabilit y of this latter ev en t in the case x ′′ j ∈ [ x ′ j , x j ] and λ ( y ) ≤ m j − 2 − m for all y ∈ [ x ′′ j , x j ] . W e ﬁrst notice Λ ( ˜ x n , ˜ h n ) ( x ′ j ) ≥ m j , and that the increments of Λ ( ˜ x n , ˜ h n ) in [ x ′ j , x j ] are ab ov e those of a Bro wnian motion, so if ( W y ) y ≥ x ′ j is a Brownian motion, P λ,χ (min y ∈ [ x ′ j ,x j ] Λ ( ˜ x n , ˜ h n ) ( y ) < m j − 2 − m − 2 ) ≤ P λ,χ (min y ∈ [ x ′ j ,x j ] ( W y − W x ′ j ) < − 2 − m − 2 ) . Thus Lemma 24 yields that TRUE SELF-REPELLING MOTION ABOVE A GENERAL BARRIER 29 when m is large enough, (4) P λ,χ min y ∈ [ x ′ j ,x j ] Λ ( ˜ x n , ˜ h n ) ( y ) < m j − 2 − m − 2 ! ≤ exp  − 5 m 2 2 m +5 v  . In addition, denoting N = inf { n ′ ∈ N | ˜ x n ′ < x ′′ j , Λ ( ˜ x n ′ , ˜ h n ′ ) ( x ′′ j ) = λ ( x ′′ j ) } (which may b e inﬁnite), we ha v e P λ,χ ( ∃ n ′ ∈ N , Λ ( ˜ x n ′ , ˜ h n ′ ) ( x ′′ j ) = λ ( x ′′ j ) , Λ ( ˜ x n ′ , ˜ h n ′ ) ( x j ) ≤ m j − 2 − m +1 ) ≤ X n ′ ∈ N , ˜ x n ′ m j − 2 − m − 1 ) . F or all y ∈ [ x ′′ j , x j ] we hav e λ ( y ) ≤ m j − 2 − m = λ ( x ′′ j ) , so b y Lemma 27 we hav e P λ,χ (max y ∈ [ x ′′ j ,x j ] | Λ ( x ′′ j ,λ ( x ′′ j )+2 − m − 2 ) ( y ) − Λ ( x ′′ j ,λ ( x ′′ j )+2 − m − 2 ) ( x j ) | ≥ 2 − m − 2 ) ≤ exp( − 5 m 2 m +5 v ) when m is large enough. Therefore, when m is large enough, (7) P λ,χ (Λ ( x ′′ j ,λ ( x ′′ j )+2 − m − 2 ) ( x j ) > m j − 2 − m − 1 ) ≤ exp  − 5 m 2 m +5 v  . Finally w e consider P λ,χ ( D ( x j , m j − 2 − m +1 , m j + 2 − m +2 , 2 − m 3 )) ≤ C (2 − m +3 / p 2 − m 3 ) 3 = C 2 9 2 − 5 2 m b y Lemma 44. Along with (4), (5), (6) and (7), this yields that when m is large enough, P λ,χ ( M 3 j ∩ A 3 j ) ≤ 4 exp( − 5 m 2 2 m +5 v ) + C 2 9 2 − 5 2 m , whic h ends the pro of of Lemma 49. □ □ 30 LAURE MARÊCHÉ 6. ( λ, χ ) -true self-repelling motion: proof of the resul ts in Section 2.3 In this section w e giv e the pro ofs for the construction and properties of the ( λ, χ ) -true self-rep elling motion and its lo cal time, assuming ( λ, χ ) is a go o d barrier. The pro of of Theorem 20 is the same as that of Theorem 4.2 of [32], while the pro of of Theorem 23 stems directly from the deﬁnitions, so w e do not detail them. How ev er, Prop ositions 15, 17 and 21 need more attention, hence we will give their pro ofs. As in [32], w e b egin b y deﬁning, for an y ( x, h ) ∈ R 2 λ , D ( x, h ) = { ( x ′ , h ′ ) ∈ R 2 | λ ( x ) < h ′ ≤ ¯ Λ ( x,h ) ( x ′ ) } , and stating the follo wing result, whose proof is the same as the pro of of Prop osition 3.1 of [32]. Prop osition 50. Almost sur ely, for any ( x 1 , h 1 )  = ( x 2 , h 2 ) in R 2 λ , one of the fol lowing o c curs: • either ( x 1 , h 1 ) ∈ D ( x 1 , h 1 ) ⊂ D ( x 2 , h 2 ) , ( x 2 , h 2 ) ∈ D ( x 1 , h 1 ) and D ( x 2 , h 2 ) \ D ( x 1 , h 1 ) c ontains a non-empty op en set; • or ( x 2 , h 2 ) ∈ D ( x 2 , h 2 ) ⊂ D ( x 1 , h 1 ) , ( x 1 , h 1 ) ∈ D ( x 2 , h 2 ) and D ( x 1 , h 1 ) \ D ( x 2 , h 2 ) c ontains a non-empty op en set. Before pro ving Prop ositions 15, 17 and 21, we also need the follo wing t w o technical results, Lemmas 51 and 52. They are easy in the classical case and so w ere not prov en in [32]. Ho w ev er, they are more complicated with our more general barriers, which is wh y we show them. In particular, the ﬁrst of these results, stating all the D ( x, h ) are b ounded, is what we need the assumption of a go o d barrier for. Lemma 51. Almost sur ely, for al l ( x, h ) ∈ R 2 λ , the set D ( x, h ) is b ounde d. Pr o of of L emma 51. The idea is that if the barrier is go o d, eac h Λ ( x,h ) will reach λ at the righ t of χ , and then be absorb ed b y λ , which will b ound D ( x, h ) on the righ t, with a similar argumen t on the left. Ho w ever, since the lemma has to hold almost surely for all ( x, h ) ∈ R 2 λ , we need some care. Since ( λ, χ ) is go o d, the ev ent {∀ k, ℓ ∈ Z , if ℓ > λ ( k ) there exists y > k so that Λ ( k,ℓ ) ( y ) = λ ( y ) } is almost sure. Moreo ver, on this ev ent, for any ( x, h ) ∈ R 2 λ , w e can c hoose an in teger k > max( χ, x ) , and an in teger ℓ > Λ ( x,h ) ( k ) . There exists y > k so that Λ ( k,ℓ ) ( y ) = λ ( y ) , hence Λ ( x,h ) ( y ) = λ ( y ) , th us Λ ( x,h ) ( z ) = λ ( z ) for all z ≥ y . Since λ and Λ ( x,h ) are con tin uous, { ( x ′ , h ′ ) ∈ R 2 | x ′ ≥ x, λ ( x ′ ) < h ′ ≤ Λ ( x,h ) ( x ′ ) } is b ounded. { ( x ′ , h ′ ) ∈ R 2 | x ′ ≤ x, λ ( x ′ ) < h ′ ≤ Λ ∗ ( x,h ) ( x ′ ) } is b ounded b y a similar argument, which ends the pro of. □ W e recall that T ( x, h ) = R + ∞ −∞ ( ¯ Λ ( x,h ) ( y ) − λ ( y ))d y . Our second tec hnical lemma is as follows. Lemma 52. Almost sur ely, for al l x ∈ R , we have lim h → + ∞ T ( x, h ) = + ∞ . Pr o of. The idea is that when h is large, Λ ( x,h ) will start so muc h ab ov e λ that T ( x, h ) will b e large, but since this has to hold almost surely for all x ∈ R , we again need to be careful. F or any m ∈ Z , K ∈ N ∗ , b y Lemma 27, when h is large enough, P (max y ∈ [ m,m +1] λ ( y ) ≤ K , inf y ∈ [ m,m +1] Λ ( m,h ) ( y ) ≤ 2 K ) ≤ 2 √ 2 v ( h − 2 K ) √ π exp( − ( h − 2 K ) 2 2 v ) whic h tends to 0 when h tends to + ∞ , hence P (max y ∈ [ m,m +1] λ ( y ) ≤ K, ∀ h ∈ N ∩ [2 K , + ∞ ) , inf y ∈ [ m,m +1] Λ ( m,h ) ( y ) ≤ 2 K ) = 0 . Moreo v er, since λ is contin uous, almost surely there exists K m ∈ N ∗ so that sup y ∈ [ m,m +1] λ ( y ) ≤ K m . Then almost surely , for an y x ∈ R , for an y integer K ≥ K ⌊ x ⌋ there exists h ∈ N ∩ [2 K , + ∞ ) so that inf y ∈ [ m,m +1] Λ ( m,h ) ( y ) ≥ 2 K . No w, let h ′ > Λ ( m,h ) ( x ) , we then ha v e ¯ Λ ( x,h ′ ) ( y ) ≥ Λ ( m,h ) ( y ) for all y ∈ [ m, m + 1] , thus T ( x, h ′ ) ≥ K , which is enough to prov e the lemma. □ W e no w giv e the pro of of Proposition 15, which allo ws to construct the ( λ, χ ) -true self-rep elling motion. Pr o of of Pr op osition 15. The pro of of this prop osition is based on that of Lemma 3.4 of [32]. The pro of of Lemma 3.4 of [32] itself can b e directly applied in our setting, but it relies on Lemmas 3.2 and 3.3 of that work, whose TRUE SELF-REPELLING MOTION ABOVE A GENERAL BARRIER 31 pro ofs require some c hanges. The ﬁrst change is that the pro of of Lemma 3.2 of [32] uses T ( R 2 λ ) ∩ [0 , α ]  = 0 for all α > 0 , whic h is not ob vious in our setting. This can b e circumv en ted b y setting their D to ∅ if this is not the case. The second and most imp ortant problem lies in the pro of of Lemma 3.3 of [32], where an argument relies on the fact the marginals of a reﬂected Brownian motion hav e no atoms, whic h is true in their setting but not in ours, since they can hav e an atom on the barrier. This requires a non-trivial ﬁx, whic h w e explain. The arguments of [32] pro vide r, q , q ′ , r ′ ∈ D so that r < q < q ′ < r ′ , and k , k ′ ∈ N so that ˜ x k ≤ r , ˜ x k ′ ≥ r ′ , chosen as in their Equations (3.11) and (3.12). In the setting of [32], almost surely Λ ( ˜ x k , ˜ h k ) ( q )  = Λ ∗ ( ˜ x k ′ , ˜ h k ′ ) ( q ) , b ecause these tw o quantities are indep enden t and one of them is the marginal of a reﬂected Bro wnian motion, thus has no atoms. This do es not hold in our case, so w e need another argumen t. In order to ﬁnish the proof, it is actually enough to pro ve there exists almost surely a rational s ∈ [ q , q ′ ] with Λ ( ˜ x k , ˜ h k ) ( s )  = Λ ∗ ( ˜ x k ′ , ˜ h k ′ ) ( s ) . Let ℓ, ℓ ′ ∈ N so that ˜ x ℓ ≤ q ≤ q ′ ≤ ˜ x ℓ ′ . W e notice that for any rational s ∈ [ q , q ′ ] , conditionally to ( λ, χ ) , if ( ˜ x ℓ , ˜ h ℓ ) , ( ˜ x ℓ ′ , ˜ h ℓ ′ ) ∈ R 2 λ , the pro cess (Λ ( ˜ x ℓ , ˜ h ℓ ) ( y )) ˜ x ℓ ≤ y ≤ s is indep endent from Λ ∗ ( ˜ x ℓ ′ , ˜ h ℓ ′ ) ( s ) (which can b e sho wn as in [32] just b efore their Lemma 9.2), hence b y Lemma 28, P λ,χ (Λ ( ˜ x ℓ , ˜ h ℓ ) ( s ) = Λ ∗ ( ˜ x ℓ ′ , ˜ h ℓ ′ ) ( s ) > λ ( s )) = 0 . This implies P ( ∃ s ∈ [ q , q ′ ] ∩ Q , Λ ( ˜ x k , ˜ h k ) ( s ) = Λ ∗ ( ˜ x k ′ , ˜ h k ′ ) ( s ) > λ ( s )) = 0 . Consequen tly , it is enough to show that if ( ˜ x ℓ , ˜ h ℓ ) , ( ˜ x ℓ ′ , ˜ h ℓ ′ ) ∈ R 2 λ , then P λ,χ ( ∀ s ∈ [ q , q ′ ] ∩ Q , Λ ( ˜ x ℓ , ˜ h ℓ ) ( s ) = Λ ∗ ( ˜ x ℓ ′ , ˜ h ℓ ′ ) ( s ) = λ ( s )) = 0 . In the case q < χ , this comes from the con tin uit y of λ and Λ ( ˜ x ℓ , ˜ h ℓ ) , as well as from Lemma 30 which yields P λ,χ ( ∀ s ∈ [ q , min( q ′ , χ )] , Λ ( ˜ x ℓ , ˜ h ℓ ) ( s ) = λ ( s )) = 0 . If q ≥ χ , then q ′ > χ , and the result comes from Theorem 11 and Lemma 30 which yield P λ,χ ( ∀ s ∈ [max( χ, q ) , q ′ ] , Λ ∗ ( ˜ x ℓ ′ , ˜ h ℓ ′ ) ( s ) = λ ( s )) = 0 . □ W e no w giv e the pro of of Proposition 17, whic h gathers basic prop erties of the ( λ, χ ) -true self-rep elling motion. Pr o of of Pr op osition 17. T o prov e that almost surely for all t ≥ 0 we hav e H t ≥ λ ( X t ) , it is enough to notice ( X t , H t ) is in the closure of a subset of R 2 λ . The fact that almost surely all the { t ≥ 0 | X t = x } , x ∈ R are unbounded comes from Lemma 52. W e now prov e almost sure con tin uit y of ( X t , H t ) t ≥ 0 . The pro of of Prop osition 3.5 of [32] is enough to show that almost surely , for any t ≥ 0 , if ( t n ) n ∈ N is a sequence con verging to t so that ( X t n , H t n ) n ∈ N has a limit, then this limit is ( X t , H t ) . Consequently , it is enough to prov e that almost surely , for an y suc h t ≥ 0 and ( t n ) n ∈ N con v erging to t , the sequence ( X t n , H t n ) n ∈ N is b ounded. In order to do that, we notice Lemma 52 almost surely pro vides us with ( x 0 , h 0 ) ∈ R 2 λ so that T ( x 0 , h 0 ) ≥ t + 1 . Moreo v er, when n is large enough, t n ≤ t + 1 / 2 . F or such n , ( X t n , H t n ) ∈ { ( x, h ) ∈ R 2 λ , T ( x, h ) ∈ ( t n − 1 / 2 , t n + 1 / 2) } , and for any ( x, h ) ∈ R 2 λ so that T ( x, h ) ∈ ( t n − 1 / 2 , t n + 1 / 2) , w e ha ve T ( x, h ) < T ( x 0 , h 0 ) , hence Prop osition 50 yields ( x, h ) ∈ D ( x 0 , h 0 ) . This implies that when n is large enough, ( X t n , H t n ) ∈ D ( x 0 , h 0 ) , which is almost surely b ounded by Lemma 51. This ends the pro of of almost sure contin uit y of ( X t , H t ) t ≥ 0 . □ W e no w deal with the pro of of Proposition 21, whic h states basic prop erties of the local times. Pr o of of Pr op osition 21. The fact that almost surely all the L t ( x ) , t ≥ 0 , x ∈ R are ﬁnite comes from Lemma 52. The functions t 7→ L t ( x ) are non-decreasing by deﬁnition, and contin uous thanks to Proposition 50 whic h implies T ( x, . ) is strictly increasing. T o prov e that almost surely for all t ≥ 0 we ha v e H t = L t ( X t ) , w e notice that if we had H t > L t ( X t ) (the case H t < L t ( X t ) can b e dealt with in the same w ay), then b y the deﬁnition of L t and since T ( X t , . ) is strictly increasing, for h ∈ ( L t ( X t ) , H t ) we would ha v e T ( X t , h ) > t , hence b y Prop osition 50 the set P t = T ε> 0 { ( x, h ) ∈ R 2 λ , T ( x, h ) ∈ ( t − ε, t + ε ) } w ould b e b elo w ¯ Λ ( X t ,h ) , so we would hav e H t ≤ h , which is a con tradiction. □ 32 LAURE MARÊCHÉ A ckno wledgements The author wishes to thank Thomas Moun tford for his helpful comments. 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True self-repelling motion above a general barrier

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