Entrance laws for coalescing and annihilating Brownian motions

En trance la ws for coalescing and annihilating Bro wnian motions. Roger T rib e and Oleg Zab oronski Abstract Systems of instan taneously annihilating or coalescing Brownian motions on the line are considered. The extreme points of the set of entrance laws for this pro cess are shown to b e Pfaffian point processes at all times and their kernels are iden tified. W e consider a system of particles mo ving, b et ween collisions, as indep endent Bro wnian motions on R . A pair of particles up on collision instan taneously react, either annihilating each other with probability θ or coalescing into a single particle with probability 1 − θ . Separate collisions pro duce independent reactions. Thus the parameter θ ∈ [0 , 1] in terp olates b et ween the purely coalescing case and the purely annihilating case. W e will write X t for the empirical measure of the p ositions of particles at time t . A suitable state space for this system ( X t ) is M , the space of lo cally finite p oint measures on R with the top ology of v ague conv ergence. The instantaneous reactions mean that X t tak es v alues in the subset M 0 ⊆ M of simple p oin t pro cesses, that is there is at most one particle at an y p oint. The pro cess has a F eller transition densit y ( p t ( µ, dν ) : t ≥ 0) and the pro cess has a Mark ov family of laws ( E Q : Q a probability supp orted on M 0 ). F or the coalescing case the process can b e constructed from finite systems using monotonicity , that is by adding one more initial particle at a time, while the general case requires more care. The details of all these basic prop erties are, for θ = 0 , 1, in the App endix 4.1 to [5]. F or the mixed case θ ∈ (0 , 1), which arises in sev eral settings (p olymer chains, multi-v alued v oter mo dels, coalescent mo dels - s ee [2]), these basic results ab ov e still hold by rep eating the arguments for θ = 1 from [5] replacing the dualit y function used there with the general θ dualit y function (7) b elow. The system is often studied starting from a sp ecific en trance law, informally starting with a particle at ev ery x ∈ R (for example this is the case with the Arratia flo w). The in tuition is that the instan taneous reactions bring the system instan taneously in to M 0 . Ho wev er a complete description of entrance la ws requires care. See [4] where a complete classification of the entrance la ws for θ = 1 is found using duality with the contin uum v oter mo del. How ev er, the Pfaffian formulae for the intensities of the one dimensional marginals clarify the picture and lead to a simple description of the complete set of entrance la ws, whic h is the purp ose of this note. The usefulness of this Pfaffian prop erty is illustrated in [1] where F redholm Pfaffians are used to study empty interv als and exit measures. 1 W e recall the Pfaffian structure of X t , yielding explicit Leb esgue in tensities ρ t ( x 1 , . . . , x n ) for X t . These were studied first for θ = 0 for the Arratia flow, and the analogous flow for θ = 1, in [5], and then more generally in [2] and [3] for mixed systems, together with asso ciated systems where certain branc hing or immigration mec hanisms are presen t. Started from a deterministic initial condition µ ∈ M 0 the point pro cess X t is, at an y fixed time t > 0, a Pfaffian point pro cess. This means that its intensities are giv en in terms of a Pfaffian ρ t ( x 1 , . . . , x n ) = pf ( K t ( x i , x j )) for x 1 < x 2 < . . . < x n (1) where the k ernel K t : V 2 → M 2 × 2 ( R ), where V 2 = { ( x, y ) : x < y } , is constructed from the initial condition µ as w e now recall. Firstly K t is in ’derived form’, that it is deriv ed from a scalar k ernel K t : V 2 → R via the relation K t ( x, y ) = 1 1 + θ  K t ( x, y ) − D x K t ( x, y ) − D y K t ( x, y ) D xy K t ( x, y )  for t > 0 and x < y , (2) and K t ( x, x ) is skew-symmetric with K t ( x, x ) 1 , 2 = − 1 1+ θ D x K t ( x, x ). The scalar kernel K ∈ C 1 , 2 ((0 , ∞ ) × V 2 ) is the unique b ounded solution to the heat equation  ∂ t K = ∆ K on (0 , ∞ ) × V 2 , K t ( x, x ) = 1 for x ∈ R , (3) satisfying the initial condition K t → K 0 in distribution as t ↓ 0 on V 2 , where K 0 ( x, y ) = ( − θ ) X 0 ( x,y ) for ( x, y ) ∈ V 2 . (4) The pro of of this Pfaffian structure is based on the Marko v dualit y formula E [( − θ ) X t ( x 1 ,x 2 )+ X t ( x 3 ,x 4 )+ ... + X t ( x 2 n − 1 ,x 2 n ) ] = pf ( K t ( x i , x j ) : i, j ≤ 2 n ) . (5) F or coalescing-annihilating random walks this is Lemma 7 of [2]. Expression (5) is the cor- resp onding contin uous limit obtained by follo wing the argumen ts of Section 3 of that pap er. Here, and throughout, w e write µ ( a, b ) (and µ ( a, b ] e.t.c.) as shorthand for µ (( a, b )). W e also use the con ven tion that 0 k = I ( k = 0), so that for instance when θ = 0 the expression ( − θ ) X 0 ( x,y ) b ecomes the indicator I ( X 0 ( x, y ) = 0) that there are no particles inside ( x, y ). W rite ( T t ) for the Mark ov semigroup of ( X t ) acting on b ounded measurable F : M 0 → R . Recall that an en trance law for ( X t ) is a family of laws ( Q t : t > 0) on M 0 so that Z M 0 T t F ( µ ) Q s ( dµ ) = Z M 0 F ( µ ) Q t + s ( dµ ) for all s, t > 0 and all F . (6) Clearly , en trance la ws form a con vex set and our aim is classify the extreme p oints of this set. Notation. W e write ( K f t ) for the solution to the heat equation (3) with initial condition K f 0 = f . W e write Q f = ( Q f t : t > 0) for the family of laws on M 0 , when it exists, where Q f t 2 is the law of the p oint pro cess with intensities ρ ( n ) t giv en via (1) for the kernels K f t arising as in (2) from the scalar kernel K f t . As explained ab ov e, starting from a deterministic condition, the law of ( X t : t > 0) is giv en b y Q f with f ( x, y ) = ( − θ ) X 0 ( x,y ) . The aim of this note is to show that all en trance laws are mixtures of Q f for suitable functions f : V 2 → R . Theorem 1 The extr eme elements of the set of entr anc e laws for ( X t ) ar e ( Q f : f ∈ C θ ) wher e C θ ⊆ L ∞ ( V 2 ) is given by C 1 = { f ( x ) f ( y ) , ( x, y ) ∈ V 2 : me asur able f : R → [ − 1 , 1] } , and for θ ∈ [0 , 1) C θ = { I ( S ∩ ( x, y ) = ∅ ) , ( x, y ) ∈ V 2 : close d S ⊆ R } . Mor e over these laws Q f ar e distinct, that is f  = g implies Q f  = Q g . The dualit y function used to analyse mixed systems is, for θ ∈ [0 , 1] and µ ∈ M 0 , s µ ( x, y ) := ( − θ ) µ ( x,y ) for ( x, y ) ∈ V 2 . (7) Here is the k ey underlying lemma, whose pro of is dela yed un til the end of this note. Lemma 2 The we ak- ∗ closur e in L ∞ ( V 2 ) , as dual to L 1 ( V 2 ) , of the set S θ = ( s µ ( x, y ) , ( x, y ) ∈ V 2 : µ = n X k =1 δ x k ∈ M 0 , n ≥ 0 ) of finite spin functions is ˜ C θ , wher e ˜ C 1 = C 1 and for θ ∈ [0 , 1) it is the set ˜ C θ = n I ( S c ∩ ( x, y ) = ∅ )( − θ ) P z ∈ S i ∩ ( x,y ) w ( z ) : close d S ⊆ R , w : S i → N ∪ {∞} o wher e S = S i ∪ S c is the disjoint de c omp osition of a close d set S into its isolate d p oints S i and its cluster p oints S c . Remarks. 1. The sup erset ˜ C θ ⊇ C θ , when θ ∈ [0 , 1), will lab el entrance la ws (via the map f → Q f ); ho wev er only the set C θ will lab el extremal en trance laws. 2. All the functions here lie in the unit ball B 1 = { f : V 2 → [ − 1 , 1] } and the weak- ∗ topology is metrizable on this ball (since L 1 ( V 2 ) is separable). Pro of of Theorem. W e first c heck that Q f , for f ∈ ˜ C θ , do form entrance la ws. W e follow the steps from [5] where an entrance la w for the cases θ ∈ { 0 , 1 } when f ≡ 0 w as constructed 3 (called there a ’maximal’ en trance law and informally corresp onding to starting a particle at ev ery p oint in R as for the Arratia flo w). Fix f ∈ ˜ C θ . By Lemma 2 we can c ho ose a sequence ( µ n ) so that s µ n → f weak- ∗ . Let ( X ( n ) t ) b e the corresp onding particle system with the initial condition µ n . At a fixed t > 0, the corresp onding scalar k ernels K ( n ) t , solving (3) with initial condition K ( n ) 0 = s µ n , are giv en, for ( x, y ) ∈ V 2 , b y K ( n ) t ( x, y ) = 1 + Z V 2 ( g t ( x − x ′ , y − y ′ ) − g t ( y − x ′ , x − y ′ ))( s µ n ( x ′ , y ′ ) − 1) dx ′ dy ′ (8) where g t ( x, y ) = (1 / 4 π t ) exp( − ( x 2 + y 2 ) / 4 t ), x, y ∈ R 2 . Using this one sees that K ( n ) t , together with their deriv atives D x K ( n ) t , D y K ( n ) t , D xy K ( n ) t , con verge b ounded p oint wise to K f t and its asso ciated deriv ativ es. This con vergence of the kernels K ( n ) t implies that the asso ciated Pfaffian p oin t pro cesses X ( n ) t con verge in law to a limiting p oint pro cess X t with law Q f t (see Lemma 10 in [2]). Moreo ver the Marko v dualit y formula extends to hold for the limit, that is when X t has la w Q f t then E [( − θ ) X t ( x 1 ,x 2 )+ X t ( x 3 ,x 4 )+ ... + X t ( x 2 n − 1 ,x 2 n ) ] = pf ( K f t ( x i , x j ) : i, j ≤ 2 n ) (9) when t > 0 , x 1 < x 2 < . . . < x 2 n . Note that this formula implies the distinctness claimed in the theorem; if f  = g then (at least for small t > 0) the kernels K f t and K g t will b e distinct and the dualit y formula sho ws that the laws Q f and Q g are not equal. P assing to the limit n → ∞ in the semigroup prop ert y for ( X ( n ) t ) for b ounded con tinuous F Z M 0 T t F ( µ ) P [ X ( n ) s ∈ dµ ] = Z M 0 F ( µ ) P [ X ( n ) t + s ∈ dµ ] , and using the F eller prop erty to see that T t F is still contin uous, we see that ( Q f t : t > 0) satisfies the en trance law equation (6), finishing the pro of that it is an en trance law. W e now fix an y entrance la w ( Q t : t > 0) and show that it is a mixture of the en trance laws ( Q f t : t > 0) constructed ab ov e. The en trance law equation (6) giv es for any 0 < r < t Z M 0 F ( µ ) Q t ( dµ ) = Z M 0 T t − r F ( µ ) Q r ( dµ ) = Z B 1 Z M 0 F ( ν ) Q f t − r ( dν ) Q r ( { µ : s µ ∈ d f } ) (10) where w e ha ve used the fact that started from µ the law of X t − r is Q f t − r for f = s µ . The pushforw ard la w Q r ( { µ : s µ ∈ d f } ) on L ∞ ( V 2 ) is supp orted on the unit ball B 1 ⊆ L ∞ ( V 2 ) since | s µ | ≤ 1. The unit ball B 1 is w eak- ∗ compact b y Alaoglu’s Theorem (and metrizable as remark ed ab o ve). Th us the space of probabilit y measures on B 1 is itself metrizable and compact (using weak con vergence of measures) and there is a sequence r n → 0 along which the limit Θ( d f ) := lim n →∞ Q r n ( { µ : s µ ∈ d f } ) 4 exists. Since Q r n is supp orted on ˜ C θ whic h is weak- ∗ closed, the limit measure Θ is supp orted on ˜ C θ . W e no w aim to pass to the limit r n ↓ 0 in (10) to produce Z M 0 F ( µ ) Q t ( dµ ) = Z ˜ C θ Z M 0 F ( ν ) Q f t ( dν )Θ( d f ) (11) This will b e true pro vided b oth (i) f 7→ R M 0 F ( ν ) Q f t ( dν ) is a bounded con tinuous function and (ii) R M 0 F ( ν ) Q f t − r ( dν ) 7→ R M 0 F ( ν ) Q f t ( dν ) uniformly ov er B 1 as r ↓ 0. W e will c hec k (i) and (ii) for a la w determining set of functions F , which implies that Q t ( dν ) = Z ˜ C θ Q f t ( dν )Θ( d f ) (12) sho wing that any entrance la w is a mixture as w e hop ed. W e choose, for our choice of function F in (11), a Laplace functional defined, when ν = P i δ x i , by F ϕ ( ν ) = Q i (1 − ϕ ( x i )), for ϕ : R → [0 , 1) con tinuous and compactly supp orted. Then R F ( ν ) Q f t ( dν ) will b e a F redholm Pfaffian, indeed Z M 0 F ϕ ( µ ) Q f t ( dµ ) = 1 + ∞ X k =1 ( − 1) k k ! Z R k k Y i =1 ϕ ( x i ) pf ( K f t ( x i , x j ) : i, j ≤ k ) dx 1 . . . dx k . Eac h term in the series is a contin uous function of f ∈ B 1 (see the explicit formula for K t ( x, y ) in (8)) and the infinite series can b e controlled using the Hadamard bound on determinants: | pf ( K t ( x i , x j ) : i, j ≤ k ) | = | det( K t ( x i , x j ) : i, j ≤ k ) | 1 / 2 ≤ ∥ K t ∥ k ∞ (2 k ) k/ 2 and that K f t and its deriv ativ es are all b ounded functions at t > 0. Similar estimates, using the b oundedness of time deriv atives at t > 0, establish the conv ergence in condition (ii) ab ov e. The representation (12) is not quite a Cho quet representation for elemen ts of a conv ex set, but it remains to iden tify the extremal elements of ( Q f : f ∈ ˜ C θ ). W e first explain the in tuition of wh y , when θ ∈ [0 , 1) and k ≥ 2, the en trance la w Q f k corresp onding to f k ( x, y ) = ( − θ ) k I ( x< 0 ϵ then I ( S ∩ ( x, y ) = ∅ ) = 1 for almost all z − ϵ < x < y < z + ϵ which implies I ( S 1 ∩ ( x, y ) = ∅ ) = I ( S 1 ∩ ( x, y ) = ∅ ) = 1 for almost all z − ϵ < x < y < z + ϵ , and th us that z ∈ S c 1 ∩ S c 2 . W e conclude that S = S 1 = S 2 as desired. Finally w e chec k that the en trance laws corresponding to f ∈ ˜ C θ whic h hav e at least one isolated p oin t a with w eight w ( a ) ≥ 2 are not extreme. As suggested ab ov e, we claim that if f k ( x, y ) = I ( S c ∩ ( x, y ) = ∅ )( − θ ) P z ∈ S i ∩ ( x,y ) w ( z ) where w ( a ) = k ≥ 2 then Q f k = p k Q f 1 + (1 − p k ) Q f 0 . It is enough to c heck that the duality formulae (9) add up correctly at all t > 0, since these form ulae determine the la ws of a simple p oint pro cess at a fixed time t > 0. W e use that the exp ectations u (2 n ) t ( x 1 , . . . , x 2 n ) = E [( − θ ) X t ( x 1 ,x 2 )+ X t ( x 3 ,x 4 )+ ... + X t ( x 2 n − 1 ,x 2 n ) ] are the unique b ounded solutions to the linear p.d.e.’s: for n ≥ 1 ∂ t u (2 n ) t ( x ) = 1 2 ∆ u (2 n ) t ( x ) when x 1 < x 2 < . . . < x 2 n , u (2 n ) ( x ) = u (2 n − 2) t ( x \ { x k , x k +1 } ) when x 1 < . . . x k = x k +1 < . . . < x 2 n , supplemen ted with the appropriate initial condition. The equation and the b oundary condi- tions follow from those for the n = 2 case (3) and the Pfaffian formula (9) for u (2 n ) t . Thus w e need only c heck that the initial conditions for these p.d.e.’s add up, that is f k ( x 1 , x 2 ) . . . f k ( x 2 n − 1 , x 2 n ) = p k f 0 ( x 1 , x 2 ) . . . f k ( x 2 n − 1 , x 2 n )+(1 − p k ) f 1 ( x 1 , x 2 ) . . . f k ( x 2 n − 1 , x 2 n ) . Ho wev er this just reduces to the simple identit y ( − θ ) k = p k + (1 − p k )( − θ ). 6 Pro of of Lemma 2. W e argue separately for the cases θ = 1 and θ ∈ [0 , 1). W e will show (i) the closure of S θ m ust contain ˜ C θ and (ii) that ˜ C θ are w eak- ∗ closed. Case θ = 1 . In this case, for µ ∈ M 0 , the spin function factorises s µ ( x, y ) = ˆ s µ ( x ) ˆ s µ ( y ), for Leb esgue almost all ( x, y ), where ˆ s µ ( x ) =  ( − 1) µ (0 ,x ) for x ≥ 0, ( − 1) µ ( x, 0) for x < 0. Indeed the factorization is an equalit y pro vided x and y a void the supp ort of µ . Fix measurable f : R → [ − 1 , 1]. W e will construct b elow a sequence ( µ n : n ≥ 1) of finite simple p oint measures so that ˆ s µ n → f using w eak- ∗ conv ergence in L ∞ ( R ). W e write f ⊗ g for the function defined b y f ⊗ g ( x, y ) := f ( x ) g ( y ). Then the factorisation implies that s µ n → f ⊗ f using weak- ∗ con vergence in L ∞ ( V 2 ) or in L ∞ ( R 2 ). Thus the w eak- ∗ closure of S θ con tains ˜ C 1 . Define an appro ximation f n to f , for n ≥ 1, b y f n ( x ) = 1 for x ∈ [ − n, n ) and f n ( x ) =  1 for x ∈ [ k /n, a k,n ), − 1 for x ∈ [ a k,n , ( k + 1) /n ), for k = − n 2 , . . . , n 2 − 1 where a k,n are chosen so that R ( k +1) /n k/n f ( x ) dx = R ( k +1) /n k/n f n ( x ) dx . Note f n = ˆ s µ n (almost ev erywhere) for a finite measure µ n ∈ M 0 . Fix ϕ ∈ L 1 ( R ) and ϵ > 0. Cho ose ˜ ϕ smo oth compactly supp orted so that ∥ ϕ − ˜ ϕ ∥ L 1 ≤ ϵ . Then, if ˜ ϕ is supp orted in [ − L, L ], | ( f n − f , ϕ ) | ≤ | ( f n − f , ˜ ϕ ) | + | ( f n − f , ˜ ϕ − ϕ ) | ≤ | ( f n − f , ˜ ϕ ) | + 2 ϵ = X k | Z ( k +1) /n k/n ( f n ( x ) − f ( x ))( ˜ ϕ ( x ) − ˜ ϕ ( k /n )) dx | + 2 ϵ ≤ 2( L + 1) n n 2 ∥ ˜ ϕ ′ ∥ ∞ + 2 ϵ. This establishes the desired weak- ∗ conv ergence ˆ s µ n = f n → f . T o sho w that ˜ C 1 is closed we supp ose f n ⊗ f n → ψ weak- ∗ in L ∞ ( V 2 ) as n → ∞ . By weak- ∗ compactness of the unit ball, w e may choose a subsequence n k where f n k → f ∞ w eak- ∗ in L ∞ ( R ) as k → ∞ . Then f n k ⊗ f n k → f ∞ ⊗ f ∞ w eak- ∗ in L ∞ ( R 2 ), and this iden tifies the limit p oin t ψ in the pro duct form ˜ ψ = f ∞ ⊗ f ∞ as desired. Case θ ∈ [0 , 1) . Let M s b e the set of s -finite counting measures, that is µ ∈ M s if it is a coun table sum of finite p oint measures on R . W e will b elow iden tify the weak- ∗ closure of S θ as ˜ C θ = { s µ ( x, y ) : µ ∈ M s } (15) where s µ is still defined as in (7), understanding that ( − θ ) ∞ = 0. The representation (15) of a limit p oints is not unique; but the function s µ is uniquely identified via the closed supp ort 7 S = S c ∪ S i of µ and the mass µ ( { a } ) of any isolated p oin t a ∈ S i . Indeed, noting that the isolated p oin ts S i are lo cally finite, for ( x, y ) ∈ V 2 , s µ ( x, y ) = ( − θ ) µ ( x,y ) = ( 0 if ( x, y ) ∩ S c  = ∅ , ( − θ ) P a ∈ S i ∩ ( x,y ) µ ( { a } ) if ( x, y ) ∩ S c  = ∅ . Th us the s et agrees with the formula for ˜ C θ giv en in Lemm a 2 when w ( a ) = µ ( { a } ). Moreov er, the new formula, indexed b y S and w , gives distinct functions s µ ( x, y ) in L 2 ( V 2 ), so that this iden tification is bijective. T o establish (15) w e ma y write an y µ ∈ M s as µ = P ∞ k =1 δ x k , where the positions ( x k ) are not necessarily disjoin t. W e can find P ∞ k =1 δ x k,n where the sequences ( x k,n : k ≥ 1) do ha ve disjoin t elemen ts and where, for each k ≥ 1 we hav e lim n →∞ x k,n = x k . W e set µ n = P n k =1 δ x k,n . Then s µ n → s µ w eak- ∗ sho wing that ˜ C θ is contained in the weak- ∗ closure. T o see that the set ˜ C θ is itself weak- ∗ closed we supp ose that µ n ∈ M s and s µ n → ψ weak- ∗ . F or eac h n the measure µ n can b e written as µ n = P ∞ k =1 µ k,n where each µ k,n is either zero or a single p oin t mass supported in [ − k , k ]. Then b y diagonalisation w e can find a sub-sequence n ′ along whic h the measures µ k,n ′ are weakly con vergen t for all k ≥ 1. Set µ k, ∞ = lim n ′ →∞ µ k,n ′ and µ ∞ = P k µ k, ∞ ∈ M s . Then s µ n ′ → s µ ∞ w eak- ∗ , confirming that ˜ C θ is closed. References [1] W. FitzGerald, R. T rib e, and O. Zab oronski. Asymptotic expansions for a class of Fredholm Pfaffians and in teracting particle systems. The Annals of Pr ob ability , 50(6):2409–2474, 2022. [2] B. 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