Survival probability of mutually killing Brownian motions and the OConnell process

Surviv al probabili t y of m utually kil ling Bro wnian motions and the O’Connell pro cess Mak oto Kator i ∗ 23 March 2012 Abstract Recen tly O’Connell in tro duced an in teracting diffusive particle system in order to study a d irected p olymer mo del in 1+1 d im en sions. The in finitesimal generator of the pro cess is a harmonic transform of the quan tum T o da-lattice Hamiltonian by the Whittak er function. As a ph ysical in terpretation of this construction, w e sho w that the O’Connell pro cess without drift is realized as a system of m utu ally killing Bro wnian motions conditioned that all particles survive forev er. When the c h aracteristic length of in teractio n killing other p articles go es to zero, the p ro cess is red u ced to the n on colliding Bro w nian motion (the Dyson mo del). Keyw ords Mutu ally killing Brownian motions · Sur viv al prob ab ility · Quantum T o da lattice · Whittak er functions · The Dyson m o del 1 In tro ductio n In this paper we in tro duce a sys tem of finite num b er of o ne- dimensional Bro wnian motions whic h kill eac h other, and ev a lua te long-t erm asymptotics o f the probabilit y that all particles surviv e. Then w e define a pro ce ss of m utually killing Bro wnian motions c onditione d that al l p articles survive for ever . W e sho w t ha t this conditional pro ces s is equiv alen t to a sp ecial case of the pro cess recen tly in t ro duced b y O ’Connell in order to analyze a directed p olymer mo del in 1+1 dimensions [28]. As an in tro duction of our study of many-particle syste ms, here w e consider a family of one-particle systems with a pa r ameter ξ > 0. Let B ( t ) , t ≥ 0 b e suc h a one-dimensional Bro wnian motion that its surviv al pro ba bilit y P ( t ) deca ys f ollo wing the equation dP ( t ) dt = − V ( B ( t )) P ( t ) , t ≥ 0 , (1.1) ∗ Department of Physics, F aculty of Science and Engineering, Chuo Universit y , Kasug a, Bunkyo-ku, T okyo 112-8 551, Japa n; e-ma il: k atori@ phys.ch uo - u.ac.jp 1 with a deca y rate function V ( x ) = 1 2 ξ 2 e − 2 x/ξ , x ∈ R . (1.2) The function (1.2) implies that, if the Brownian pa r t icle mov es in the p ositiv e region far from the origin x ≫ ξ , deca y of surviv al probability is negligible, while as it a ppro ac hes the origin the deca y b ecomes large. Note that the Bro wnian particle is able to p ene trate a negativ e region x < 0, but there the deca y rate of surviv al probability grows exp onentially as a function of | x | . The para meter ξ is the char acteristic length of the inter ac tion to kill a particle acting from t he origin and it is also the p en e tr ation le n gth of a part icle in to the negativ e region. See Fig.1(a). F or 0 < t < ∞ , if a path of the Brownian motion up to time t is giv en as { B ( s ) : 0 ≤ s ≤ t } , the surviv al probability of the particle at the time t is giv en b y an in tegration of (1.1), P ( t |{ B ( s ) : 0 ≤ s ≤ t } ) = exp  − Z t 0 V ( B ( s )) ds  = exp  − 1 2 ξ 2 Z t 0 e − 2 B ( s ) /ξ ds  . (1.3) Pro vided that B (0) > 0, in t he limit ξ → 0 , the pro ces s b ecomes a n absorbing Brow nian motion in the p ositiv e r egion with an absorbing w all at t he origin, in whic h the surviv al probabilit y (1.3) is zero if the particle hits the origin at an y time s ≤ t , and it is one if it sta ys in the p ositiv e region in the time p eriod [0 , t ]. See Fig.1(b). Let Q ( t, y | x ) b e the tra nsition probability densit y o f this killing Brownian motion from a p osition x ∈ R to a p osition y ∈ R during t ime t > 0. It will b e obtained b y ta king an a v erag e of (1.3 ) ov er all realizations of Brow nian path under the conditions B (0) = y , B ( t ) = x . (Note that the time ordering is rev ersed, since the equation of Q giv en by (1.5) b elo w is the b ackwar d Kolmog o ro v equation.) That is, Q ( t, y | x ) = E  exp  − Z t 0 V ( B ( s )) ds  1 ( B (0) = y , B ( t ) = x )  , (1.4) where E [ · ] denotes an expectatio n with respect to Bro wnian motion, and 1 ( ω ) is the indicator function of a condition ω ; 1 ( ω ) = 1 if ω is satisfied, and 1 ( ω ) = 0 otherwise. It is the F eynman-Kac formula (see, e.g. , [15]) for the unique solution of the diffusion equation ∂ ∂ t Q ( t, y | x ) =  1 2 ∂ 2 ∂ x 2 − V ( x )  Q ( t, y | x ) (1.5) under the initial condition Q (0 , y | x ) = δ ( x − y ). Let I ν ( z ) , z , ν ∈ C b e the mo dified Bessel function of the first kind I ν ( z ) = ∞ X k =0 ( z / 2 ) ν +2 k Γ( k + 1)Γ( k + ν + 1) , | z | < ∞ , | a r g z | < π , (1.6) 2 t x x 0 ξ ( b ) ( a ) t 0 ξ Figure 1: (a ) An illustrat io n of a path of surviving Brownian particle in one-dimension with a killing term − V ( x ) giv en by (1.2). The para meter ξ is the ch aracteristic length of the in teraction t o kill a par t icle a cting from the o r ig in. It also indicates the p enetration length of a particle in t o the nega t ive region. (b) An illustratio n of a path of surviving Brow nian particle in the absorbing Brownian motion with an absorbing w all at the origin. It is regar ded as the limit ξ → 0 of the system shown in (a). and K ν ( z ) , z ∈ C b e Macdonald’s function [3 7 , 25] defined b y K ν ( z ) = π 2 I − ν ( z ) − I ν ( z ) sin( ν π ) , | arg z | < π , for ν 6 = 0 , ± 1 , ± 2 , . . . , (1.7) and for in tegers ν = n , K n ( z ) = lim ν → n K ν ( z ) , n = 0 , ± 1 , ± 2 , . . . , (1.8) whic h are b oth analytic functions of z fo r all z in the complex plane C cut along the negativ e real axis, and en tire functions of ν . W e see that I ν ( z ) and K ν ( z ) are linearly indep enden t solutions of the differen tial equation d 2 w dz 2 + 1 z dw dz −  1 + ν 2 z 2  w = 0 . F or x > 0 and ν ≥ 0, I ν ( x ) is a p ositiv e function whic h increases monoto nically as x → ∞ , while K ν ( x ) is a p ositiv e function which decreases monoto nically as x → ∞ . Then an in tegral represen tation of Q ( t, y | x ) is give n b y [26, 27, 28, 17] Q ( t, y | x ) = 1 π Z ∞ −∞ e − k 2 t/ 2 K iξ k ( e − x/ξ ) K − iξ k ( e − y /ξ ) | Γ( iξ k ) | − 2 dk , (1.9) where i = √ − 1 a nd Γ( z ) is t he gamma function. 3 F or eac h initial p osition x ∈ R , the surviv al probabilit y o f this killing Bro wnian motion at time T ≥ 0 is o bta ined b y in tegra ting Q ( T , y | x ) ov er y ∈ R , N ( T , x ) = Z ∞ −∞ Q ( T , y | x ) dy , x ∈ R , T ≥ 0 . (1.10) If w e condition the pro cess to surviv e up to time T > 0, the transition probabilit y densit y from the state ( s, x ) to ( t, y ) of the killing Brow nian motion is given b y P T ( s, x ; t, y ) = N ( T − t, y ) N ( T − s, x ) Q ( t − s, y | x ) , x, y ∈ R , 0 ≤ s ≤ t ≤ T . (1.11) If w e use the in tegral represen tat ions of K ν ( z ) [37, 25], K ν ( z ) = 2 ν Γ( ν + 1 / 2) z ν √ π Z ∞ 0 cos( z u ) (1 + u 2 ) ν +1 / 2 du, (1.12) and K ν ( z ) = 1 2 Z ∞ 0 s ν − 1 exp  − z 2  s + 1 s  ds, (1.13) w e can ev aluat e the long-term asymptotics of t he surviv a l probabilit y for | x | < ∞ , N ( T , x ) = C T − 1 / 2 K 0 ( e − x/ξ ) ×  1 + O  x √ T  in x √ T → 0 (1.14) with C = 3 ξ p 2 /π as sho wn in App endix A. Then w e can ta k e the limit T → ∞ of ( 1.11) as P ( t, y | x ) ≡ lim T →∞ P T (0 , x ; t, y ) = K 0 ( e − y /ξ ) K 0 ( e − x/ξ ) Q ( t, y | x ) , x, y ∈ R , 0 ≤ t < ∞ . (1.15) This is the transition probability densit y of the killing Bro wnian motion c onditione d to survive for ever . W e should note that this conditional Brow nian motion is the diffusion pro cess with infinitesimal generator L MY = 1 2 d 2 dx 2 + d dx n log K 0 ( e − x/ξ ) o d dx . (1.16) Matsumoto and Y or [26, 27] hav e studied the exp o nential functionals o f Brownian mot io n Z MY ( t ) = ξ log  1 ξ 2 Z t 0 e 2 B ( s ) /ξ ds  − B ( t ) , t ≥ 0 , (1.17) whic h is the diffusion pro cess whose infinitesimal generator is (1 .1 6). W e can sa y that the Matsumoto-Y or pro cess (1.17) is realized as the presen t killing Bro wnian motion conditioned to surviv e forev er [17]. 4 No w w e consider the limit ξ → 0 of the form ulas obtained abov e. Since I ν ( z ) is defined as the series expansion (1 .6 ), w e can see I iξ k ( e − x/ξ ) ≃ ( e − x/ξ / 2) iξ k / Γ( iξ k + 1 ) ≃ e − ik x in ξ → 0 for x > 0. Then (1.7) giv es K iξ k ( e − x/ξ ) ≃ sin ( k x ) / ( ξ k ) 1 ( x > 0) in ξ → 0. Therefore the in tegral of (1.9) can b e performed in the limit and we hav e q ( t, y | x ) ≡ lim ξ → 0 Q ( t, y | x ) = 1 √ 2 π t n e − ( x − y ) 2 / 2 t − e − ( x + y ) 2 / 2 t o 1 ( x > 0 , y ≥ 0) . (1.18) It is the transition probability densit y of the absorbing Brownian motion with an absorbing w a ll at the or ig in, whic h is easily obtained by applying the reflection principle o f Brow nian motion [15]. Applying the result (A.2) shown in App endix A, w e see that lim ξ → 0 ξ K 0 ( e − x/ξ ) = x 1 ( x > 0) , (1.19) and then (1.15) give s in this limit p ( t, y | x ) ≡ lim ξ → 0 P ( t, y | x ) = y x q ( t, y | x ) , x > 0 , y ≥ 0 , t ≥ 0 , (1.20) whic h is a harmonic transform ( h -transform) of (1.18) and is identified with the tr ansition probabilit y densit y of the three-dimensional Bes sel pro cess, BES(3) (see, for instance, [20]). As a matter of f a ct [26, 27], (1.17) giv es lim ξ → 0 Z MY ( t ) = 2 sup 0 ≤ s ≤ t B ( s ) − B ( t ) , t ≥ 0, whic h is indeed distributed as BES(3) (Pitman’s 2 M − X theorem [30]). In the presen t pap er, w e consider a n N - particle system of o ne- dimensional Brow nian motions with N ≥ 2, B ( t ) = ( B 1 ( t ) , . . . , B N ( t )) , t ≥ 0, with a p ositiv e parameter ξ > 0, suc h tha t the probability that all N particles surviv e up to time t , P N ( t ), deca ys following the equation dP N ( t ) dt = − V N ( B ( t )) P N ( t ) , t ≥ 0 (1.21) with a deca y rate function V N ( x ) = 1 ξ 2 N − 1 X j =1 e − ( x j +1 − x j ) /ξ , x ∈ R N . (1.22) W e study an integral represen tatio n of the tra nsition probability densit y Q N ( t, y | x ) , x , y ∈ R N , t ≥ 0 , whic h is a multiv ariate extension of (1.9). The extensions of the form ula s (1.14) and (1 .1 5) are shown. W e prov e that the N - particle system of the mutually killing Brow - nian motio ns following (1.21) and (1 .2 2) c ond itione d that al l N p articles survive for ever is equiv alent to a sp ecial cas e (without drift, ν = 0) of the O’Connell pro ce ss [28, 17]. In the context of quan tum mec ha nics, the deca y rate functions of surviv al probability (1.2) and (1.22) in the killing Brownian motions are considered t o g iving p oten tial energy 5 of the systems. They are called the Y ukawa p otential and the T o da- l a ttic e p otential [36], resp ectiv ely . The m ultiv a riate extensions of Macdonald’s function (1.7) with (1.8) ar e the Whittak er functions (see [1, 28, 4, 2 , 29] and references therein), whic h ha v e b een extensiv ely studied in ma t hematical ph ysics as eigenfunctions of the quan tum T o da lattice [24, 32, 31, 9, 21, 22, 23, 14, 7, 8]. See also [10]. In Sect.2, as preliminaries, useful in tegral represen tations of the Whittake r functions are giv en fo r the GL ( N , R )-quan tum T o da lattice. The transition pr o babilit y densit y of the system of m utually killing Brownian motio ns is giv en, in the case that killing o f part icles do es not o ccur during an observing time p eriod, as an inte gral of a pro duct of the Whittak er functions ov er the Sklyanin measure. Then the main results are giv en. W e also demonstrate that when the characteristic length ξ of the in teraction killing other particles go es to zero, the O’Connell pro ces s is reduced to the noncolliding Bro wnian motion, whic h is kno wn to b e equiv alent to the Dyson mo del [11 , 20] for the eigen v a lue pro cess of Hermitian matrix-v alued diffusion pro ces s ( i . e . D yson’s Brow nian motio n mo del with the parameter β = 2 [5]). The pro ofs of Lemmas and Prop osition are giv en in Sect.3. App endix is giv en for pro ving (1.14) and (1.19) used ab ov e. 2 Preliminaries and Main Results 2.1 Quan tum T o da lattice and Whittak er fun ctions In this subsection we set ξ = 1 in Eq.(1.22) and consider V N ( x ) a s a p o ten t ia l energy of a quantum N - pa rticle system in one dimension. Then w e ha v e the Hamiltonian of the GL ( N , R )-quantum T o da lattice H N = − 1 2 N X j =1 ∂ 2 ∂ x 2 j + N − 1 X j =1 e − ( x j +1 − x j ) (2.1) for N ∈ { 2 , 3 , . . . } . ( In t his pa p er, w e set the mass of particle m = 1 and ~ = 1 for simplicit y of notation.) The generalized eige n v a lue problem H N Ψ λ ( x ) = λ Ψ λ ( x ) , x ∈ R N , (2.2) is solv ed b y s etting λ = − 1 2 N X j =1 ν 2 j = − 1 2 | ν | 2 , ν = ( ν 1 , . . . , ν N ) ∈ C N (2.3) with the eigenfunction Ψ λ ( x ) = ψ ( N ) ν ( x ), where ψ ( N ) ν ( x ) is the GL ( N , R )-Whittak er function [24, 32, 31, 21] (the class-one gl N -Whittak er function [8, 1, 28, 4, 2]). When N = 2, it is expresse d b y using Macdonald’s function (1.7) as ψ (2) ( ν 1 ,ν 2 ) ( x 1 , x 2 ) = 2 e ( ν 1 + ν 2 )( x 1 + x 2 ) / 2 K ν 2 − ν 1 (2 e − ( x 2 − x 1 ) / 2 ) . 6 In this sense, the Whittake r functions { ψ ( N ) ν ( x ) } N ≥ 2 are regar ded as m ultiv ariate extensions of Macdonald’s function. Corresp onding to the tw o kinds of in tegral represen tations (1.12) and (1.13) for K ν ( z ), t w o in tegral represen tations are know n for the Whitta ker function ψ ( N ) ν ( x ) , ν ∈ C N . The in tegral expression corresponding to (1.12) is the classical one originally giv en b y Jacquet [1 3 ] and w as rewritten as the followin g form by St a de [33]. (Here w e use the notation giv en b y [21].) Let Z denote a n upp er triangular N × N matrix with unit diagonal; Z = ( Z j,k ) 1 ≤ j,k ≤ N , Z j,k =    z j,k , 1 ≤ j < k ≤ N , 1 , 1 ≤ j = k ≤ N , 0 , 1 ≤ k < j ≤ N . (2.4) W e w rite the in tegral of a function f of Z o v er a ll real Z as Z R N ( N − 1) / 2 f ( Z ) d Z ≡ N Y k =1 k − 1 Y j =1 Z ∞ −∞ dz j,k f ( Z ) . The transp ose of Z is denoted by t Z and the pr incipal minor of size j of matrix Z t Z is written as ∆ j ( Z t Z ) , 1 ≤ j ≤ N . F or x = ( x 1 , . . . , x N ) ∈ R N , ν = ( ν 1 , . . . , ν N ) ∈ C N , ψ ( N ) ν ( x ) = exp( ν · x ) Y 1 ≤ j j , and the domain of in tegra tion S is defined by the conditions min k { Im γ j,k } > max k { Im γ j +1 ,k } for a ll j = 1 , 2 , . . . , N − 1. This is called the Mellin-Barnes integral represen tation, since it can b e regarded as the m ultiv ariate extension of the Mellin-Barnes repre sen tatio n of Macdonald’s function [37 ] K ν ( z ) = 1 8 π i Z C Γ  s + ν 2  Γ  s − ν 2   z 2  − 2 ds, where the pa th of in tegration C b eing a v ertical line to the righ t of a n y p oles of the in tegrand. The deriv ation of the Mellin-Barnes in tegra l represen tation (2 .8 ) from the classical o ne (2.5) is sho wn in [34, 12] (see also [22, 23]). The equiv alence b etw een the Given tal integral rep- resen tation (2.7) and the Mellin-Barnes inte gral represen tation ( 2.8) is fully discussed in [8]. 2.2 T ransition probabilit y densit y of the m utually killing Br o wn- ian motions F or N ∈ { 2 , 3 , . . . } , w e consider the N - particle system of mutually killing Bro wnian motions, in whic h the probability that all N particles surviv e deca ys in time following (1.2 1 ) with (1.22) dep ending on realization o f pa ths. F or x , y ∈ R N , t ≥ 0, the transition probability densit y from the state x to y during time interv al t is then g iv en b y Q N ( t, y | x ) = E  exp  − Z t 0 V N ( B ( s )) ds  1 ( B (0) = y , B ( t ) = x )  = E " exp ( − 1 ξ 2 N − 1 X j =1 Z t 0 e − ( B j +1 ( s ) − B j ( s )) /ξ ds ) 1 ( B (0) = y , B ( t ) = x ) # . (2.9) It is the F eynman-Kac form ula for the solution of the diffusion equation (the bac kw ard Kolmogorov equation) ∂ u ( t, x ) ∂ t = 1 2 ∆ u ( t, x ) − V N ( x ) u ( t, x ) (2.10) with ∆ = P N j =1 ∂ 2 /∂ x 2 j under the initial condition u (0 , x ) = δ ( x − y ) ≡ N Y j =1 δ ( x j − y j ) . (2.11) 8 t x ξ ( b ) ( a ) t x ξ x 1 x 2 x 3 x 1 x 2 x 3 y 1 y 2 y 3 y 1 y 2 y 3 Figure 2: (1) An illus tration of three paths of surviving Bro wnian pa rticles in the m utually killing Bro wnian motions. The effect of killing is not negligible when the distance b etw een the nearest neigh b or pair of particles is less than ξ ; B j +1 ( t ) − B j ( t ) < ξ . The ordering of particle p ositions can b e c ha ng ed, but the risk of pair annihilation b ecomes large exp onentially , when B j ( t ) − B j +1 ( t ) > ξ . (b) In the limit ξ → 0, the presen t system con v erg es to the vicious Bro wnian motion, in whic h the pro ces s surviv es if and only if paths are not interse cting. Remark 2. If w e r ega rd (2.10) as a diffus ion equation describing an N -dimensional Br ow- nian motion in R N with a killing term − V N ( x ), (2.9) gives a transition pro babilit y densit y for a particle assumed to b e at a p osition x ∈ R N suc h that it surviv es during time t and it arriv es at a p osition y ∈ R N after t he time duration t (the F eynman-Kac fo r mula, see, for instance, [15]). In the presen t pap er, on the other hand, we w ould lik e to consider an N -part icle system of o ne-dimensional Bro wnian motions, suc h that the j -th and ( j + 1)-th particles will b e pair annihilated with high probability if B j +1 ( t ) − B j ( t ) < ξ , 1 ≤ j ≤ N − 1, and then the probability that a ll N particles surviv e av oiding fro m an y mutual killing de- ca ys in time followin g (1.2 1 ) with (1.22). In order to discuss pro ces ses, in whic h mutual killing of particles actually o ccurs and total n um b er of particles de creases in time, w e ha v e to sp ecify t he sto c hastic rule to determine which pair of particles is annihilated, when the surviv al probability conditiona l on a path, exp { − R t 0 V N ( B ( s )) ds } , b ecomes small. Here w e are in terested in, ho w ev er, the situation that m utual killing of particles do es not o ccur at all and all N part icles surviv e, follow ing the notion of vicious w alk er mo dels [6, 18, 3, 17 ]. The F eynman-Kac form ula (2.9) giv es the transition probability densit y b etw een N -pa r ticle configurations x , y ∈ R N . Figure 2(a ) illustrates three paths of surviving particles, in whic h c ha ng e of o rdering of particle p ositions o ccurs within the spatial scale ≃ ξ . By the f a ct that the Whittake r function ψ ( N ) k ( x ) solve s (2.2) with (2 .3 ), we can see that exp  − t 2 | k | 2  ψ ( N ) iξ k ( x /ξ ) , | k | 2 ≡ N X j =1 k 2 j , k ∈ R N , (2.12) 9 solv es the diffusion equation (2.10). The densit y function of the Skly anin measure [32] is defined b y s N ( k ) = 1 (2 π ) N N ! Y 1 ≤ j <ℓ ≤ N | Γ( i ( k ℓ − k j )) | − 2 (2.13) for k ∈ C N . By Euler’s reflection formula Γ( z )Γ(1 − z ) = π / sin( π z ), if k ∈ R N , Y 1 ≤ j <ℓ ≤ N | Γ( i ( k ℓ − k j )) | − 2 = Y 1 ≤ j <ℓ ≤ N  ( k ℓ − k j ) sinh π ( k ℓ − k j ) π  . Since the Whittaker functions satisfy the completeness relation with resp ect to the Skly anin measure s N ( k ) d k ≡ s N ( k ) Q N j =1 dk j [31, 21, 23, 8], Z R N ψ ( N ) i k ( x ) ψ ( N ) − i k ( y ) s N ( k ) d k = δ ( x − y ) , x , y ∈ R N , (2.14) the in tegral of the solution (2.12 ) m ultiplied b y ψ ( N ) − iξ k ( y /ξ ) on this measure, Q N ( t, y | x ) = Z R N e − t | k | 2 / 2 ψ ( N ) iξ k ( x /ξ ) ψ ( N ) − iξ k ( y /ξ ) s N ( ξ k ) d k (2.15 ) satisfies the initial condition (2.11). That is, (2 .1 5) is an in t egr a l represen tation of the transition probabilit y densit y (2.9). It should b e noted that the o rthogonality relation [2 1 , 8] Z R N ψ ( N ) − i k ( x ) ψ ( N ) i k ′ ( x ) d x = 1 s N ( k ) N ! X σ ∈S N δ ( k − σ ( k ′ )) , k , k ′ ∈ R N , (2.16) is established, where S N is a set of all p erm utation of N indices and σ ( k ′ ) ≡ ( k ′ σ (1) , . . . , k ′ σ ( N ) ). It guarantee s the Chapman-Kolmogorov equation for t he transition probability density Z R N Q N ( t 2 , z | y ) Q N ( t 1 , y | x ) d y = Q N ( t 1 + t 2 , z | x ) , x , z ∈ R N (2.17) for 0 ≤ t 1 , t 2 < ∞ , whic h should hold, since the presen t pro cess is Mark o vian. Remark 3. The completenes s relation (2.14) of the Whittak er functions is a m ultiv ariate extension of the relation for Macdonald’s functions, 1 π 2 Z R K ik ( e − x ) K − ik ( e − y ) k sinh( π k ) dk = δ ( x − y ) , x, y ∈ R , (2.18) whic h will define the Kontoro vic h-Leb edev transform [25]. W e can use (2.18) to confirm that (1.9) satisfies the initial condition Q (0 , y | x ) = δ ( x − y ). 10 2.3 Asymptotics of transition probabilit y densit y and surviv al prob- abilit y By using the integral represen tation (2.5) , we can pro v e the fo llo wing a symptotics of the transition probability densit y . Lemma 2.1 F or x , y ∈ R N , | x | < ∞ , t > 0 , as | x | / √ t → 0 , Q N ( t, y | x ) = ξ N ( N − 1) 1 (2 π ) N N !  2 t  N 2 / 2 ψ ( N ) 0 ( x /ξ ) ψ ( N ) 0 ( y /ξ ) e −| y | 2 / 2 t × Z R N e −| µ | 2 Y 1 ≤ j 0 , T → ∞ (2.35) with c 0 = p 2 /π , φ = 1 / 2. Therefore, the exp onen t φ is the same with that found in the asymptotics (1.14 ) of the surviv al pro babilit y of the killing Brownian motion discussed in Sect.1; N ( T , x ) ≃ 3 r 2 π ξ K 0 ( e − x/ξ ) T − φ , x ∈ R , T → ∞ . (2.36) If w e tak e the limit ξ → 0, (1.19) gives lim ξ → 0 N ( T , x ) ≃ cxT − φ , x > 0 , T → ∞ (2.37) with c = 3 p 2 /π = 3 c 0 . F or N ≥ 2, Proposition 2 .3 implies N N ( T , x ) ≃ C N ψ ( N ) 0 ( x /ξ ) T − φ N , x ∈ R N , T → ∞ (2.38) with C N giv en b y (2.23) and with the survival pr ob ability exp onent φ N = 1 4 N ( N − 1) (2.39) for t he presen t N - particle system of m utually killing Brow nian motion. This exp onent is the same as that for the vicious Brownian motion (the absorbing Bro wnian motion in W N ) [6, 18]; N 0 N ( T , x ) ≡ Z W N q N ( T , y | x ) d y ≃ c 0 N h N ( x ) T − φ N , x ∈ W N , T → ∞ (2 .40) with c 0 N = 2 N/ 2 Q N j =1 Γ( j / 2) π N/ 2 Q N − 1 j =1 j ! . (2.41) 14 Lemma 2.2 give s then lim ξ → 0 N N ( T , x ) ≃ c N h N ( x ) T − φ N , x ∈ W N , T → ∞ , (2.42) where c N = 2 3 N ( N − 1) / 4 π N Q N − 1 j =1 ( j !) 2 A N (2.43) with (2.24). The ab ov e results c 6 = c 0 and c N 6 = c 0 N , N ≥ 2 are consequences of the fa cts that for the transition pro babilit y densities Q ( t, y | x ) and Q N ( t, y | x ) , N ≥ 2 , the lo ng -term limit T → ∞ and the limit ξ → 0 are noncomm utable. 3 Pro ofs of Lemmas and Prop osition 3.1 Pro of of Lemma 2.1 By the in tegral represen tation (2.5), w e put ψ ( N ) iξ k ( x /ξ ) = exp ( i k · x ) b ψ ( N ) iξ k ( x /ξ ). Then b y c ha ng ing the integral v ariables as µ j = p t/ 2 k j − i ( x j − y j ) / √ 2 t, 1 ≤ j ≤ N , (2.15) with (2.13) is written as Q N ( t, y | x ) = 1 (2 π ) N N !  2 t  N/ 2 e −| x − y | 2 / 2 t × Z R N e −| µ | 2 b ψ ( N ) i α ( µ ) ( x /ξ ) b ψ ( N ) − i α ( µ ) ( y /ξ ) Y 1 ≤ j 0, K 0 ( e − β x ) is written as K 0 ( e − β x ) = 1 2 Z ∞ 0 s − 1 exp  − 1 2 e − β x  s + 1 s  ds = β 2 Z ∞ −∞ exp  − 1 2  e − β ( x − v ) + e − β ( x + v )   dv , 17 where w e ha v e set s = e β v . In the limit β → ∞ , e − β ( x − v ) + e − β ( x + v ) = 0, if v < x and v > − x , and e − β ( x − v ) + e − β ( x + v ) = ∞ , otherwise. Therefore lim β →∞ β − 1 K 0 ( e − β x ) = 1 2 Z x − x dv 1 ( x > 0) = x 1 ( x > 0) . (A.2) Then K 0 ( e − √ 2 tu/ξ ) ≃ √ 2 tu/ξ 1 ( u > 0) in t → ∞ , and I t = 2 t ξ Z ∞ 0 e − u 2 (1 + 2 u 2 ) u du × { 1 + o (1) } = 3 t ξ × { 1 + o (1) } in t → ∞ . Put this result into (A.1) , then we o btain the asymptotics (1.1 4 ). 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