OConnells process as a vicious Brownian motion

O’Connell’s pro cess as a vicious Bro wnian motion Mak oto Katori ∗ Dep artment of Physics, F aculty of Scienc e and Engine ering, Chuo Universi ty, Kasuga, Bunkyo-ku, T okyo 11 2-8551, Jap an (Dated: 10 Decem b er 2011) Abstract Vicious Br ownian motion is a diffusion scaling limit of Fisher’s vicious wa lk mo del, whic h is a system of Bro wnian particles in one dimension such that if t wo of them meet they kill eac h other. W e consider the vicious Bro wnian motion conditioned n ev er to collide with eac h other, and call it the noncolliding Bro wnian motion. Th is conditional diffusion pro cess is equiv alen t to the eigen v alue pro cess of a Hermitian-matrix-v alued Bro wn ian motion studied b y Dyson. Recen tly O’Connell in tro duced a generalization of the noncolliding Brownian motion by using th e eigenfunctions (the Whittak er functions) of the quan tum T o da lattice in order to analyze a directed p olymer mo del in 1+1 dimensions. W e consider a sys tem of on e-dimen sional Bro wnian motions with a long-ranged killing term as a generalizatio n of the vicious Bro wn ian motion and constr u ct the O’Conn ell pro cess as a conditional pr o cess of the killing Brownian motions to sur viv e forev er. P A CS num bers : 05 .40.-a,0 2.50.- r,03.65.Ge ∗ Electronic a ddress: k atori@phys.c huo-u.ac.jp 1 I. INTR ODUCTIO N Vicious w alk mo del intro duced b y Fisher [1 ] is the system of one-dimensional random w alk ers suc h that, if neighboring w alk ers meet, they kill eac h other. Though the mo del lo oks sinister, what w e are interes ted in is t o ev aluate the probability that for a finite time- in terv al any neigh b o ring pair of vicious w alk ers do not meet and thus all w alk ers surviv e; in other w ords, the probability that the p eace is kept [2–4]. If w e tak e appropriate contin uum limit (the diffusion scaling limit), w e obtain “vicious Br ownian motion” (vicious BM) [5, 6]. Assume that the num b er of particles of vicious BM is N ≥ 2 and write the p ositions as x j , 1 ≤ j ≤ N . Then the configuration space of them conditioned neve r to collide is W N = { x = ( x 1 , x 2 , . . . , x N ) ∈ R N : x 1 < x 2 < · · · < x N } , (1) whic h is called the W eyl ch am b er of type A N − 1 in the represe n tation theory [7]. The b ound- aries o f this region ∂ W N in the N -dimensional real space R N consists of the h yp erplanes x j = x j +1 , 1 ≤ j ≤ N − 1 , eac h of whic h corresp onds to o ccurrence of collision of the j -th and ( j + 1)-th particles in the vicious BM. If w e regard x a s a p osition v ector of the N - dimensional BM within W N and ∂ W N as a n absorbing b oundary suc h tha t when a par t icle hit the b oundary it is immediately absorb ed, the system is iden tified with the absorbing BM in W N . The har monic function, ∆ h N ( x ) ≡ P N j =1 ∂ 2 h N ( x ) /∂ x 2 j = 0 , x ∈ W N , satisfying the Diric hlet b oundary condition h N ( x ) = 0 , x ∈ ∂ W N , is uniquely determined up to a constant factor as h N ( x ) = Y 1 ≤ j 0 fo r t he case ~ µ = 0 and tak e the limit ε → 0 , then the limit pro cess is equiv alen t to X ( t ) , t ≥ 0. He sho w ed that the pro cess Z ~ µ ( t ) , t ≥ 0 is asso ciated with the quantum T o da lattice with t he Hamiltonian [20–22] H N = − 1 2 N X j =1 ∂ 2 ∂ x 2 j + N − 1 X j =1 e − ( x j +1 − x j ) . (9) The O’Connell pro cess is v ery r ich in mathematics connecting with quan tum in tegrable systems , repres en tation theory of Lie groups/algebras, the Whittak er functions, the ory of in tert wining r elat io ns of Mark o v pro cesses, and so on. He discussed the imp orta nce of his pro cess to study a mo del o f 1+1 dimensional directed p olymers in random environmen t with finite temp erature [19]. The purp ose of the presen t paper is to discuss the O’Connell pro cess as a generalized v ersion of vicious BM with appropriate conditions at least for the sp ecial case ~ µ = 0. (See [23–38] for other generalizations and recen t topics of vicious BM a nd noncolliding BM. W e note that in teresting connections b et w een random growth mo dels and the T o da lattice Hamiltonian is discussed in [39].) The pap er is org a nized as follo ws. In Sec.I I throug h t he F eynman-Kac formula, w e in tro duce a sys tem of Bro wnian part icles with the killing term whic h is in the same form as the p otential term in the quan tum T o da lattice Hamiltonian (9) and discuss it as a generalization of t he vicious BM. In Sec.I I I the transition proba bilit y densit y Q N ( t, y | x ) of the N -particle system of killing BMs is expressed as an integral of a pro duct of eigenfunctions of the quan tum T o da lattice ov er the Skly anin measure. Then asymptotics of Q N ( t, y | x ) in t → ∞ is estimated (Lemma 1). In Sec.IV w e introduce a drift ~ µ in our N -particle sys tem of killing BMs and define the transition probabilit y densit y of the killing BMs conditioned to surviv e up to time 0 < T < ∞ . By taking the double limits T → ∞ a nd ~ µ → 0, w e obtain the transition probability densit y P N ( t, y | x ) for the killing BMs with ~ µ = 0 conditioned to surviv e f o rev er. Th e main theorem is giv en there (Theorem 2), by whic h the 4 equiv alence b etw een the presen t conditional pro cess and the O’Connell pro cess with ~ µ = 0 is concluded. W e discuss a one-dimensional diffusion pro cess studied b y Matsumoto and Y or [40, 41] in Sec.V as a motio n of relativ e co o rdinate in the N = 2 case of our pro cess. The Matsumoto-Y or pro cess with µ = 0 is realized as a one-dimensional killing BM conditioned to surviv e fo rev er. In Sec.VI we discus s some distributions obtained by setting the sp ecial initial conditions. Sec tion VI I is devoted to summary and concluding remarks. App endix A is given for pro ving an asymptotics used in Sec.V. Some details of the N = 2 case of the O’Connell pro cess ar e giv en in App endix B. I I. QUANTUM TODA LA TTICE AND FEYNMAN-KA C F ORMULA Let N ∈ { 2 , 3 , . . . } . Consider t he eigen v alue problem o f t he quantum T o da lat tice Hamil- tonian (9), H N Ψ γ ( x ) = γ Ψ γ ( x ) , x ∈ R N . (10) F or ~ λ = ( λ 1 , . . . , λ N ) ∈ C N , t he eigenfunctions of (10) with eigenv alues γ = − 1 2 N X j =1 λ 2 j (11) ha v e b een extensiv ely studied [20–22], whic h are express ed b y ψ ( N ) ~ λ ( x ) in the presen t pap er. Let T denote a triangular array with size N , T = ( T k ,j , 1 ≤ j ≤ k ≤ N ). W e consider that the N ( N − 1 ) / 2 elemen t s T k ,j of T are indep enden t v ariables and in tro duce a f unction of them as F ( N ) ~ λ ( T ) = N X k =1 λ k k X j =1 T k ,j − k − 1 X j =1 T k − 1 ,j ! − N − 1 X k =1 k X j =1 n e − ( T k,j − T k +1 ,j ) + e − ( T k +1 ,j + 1 − T k,j ) o , (12) whic h dep ends on ~ λ = ( λ 1 , . . . , λ N ). F or a g iv en x ∈ R N , let Γ N ( x ) b e the space of a ll real triangular arrays T with size N conditioned T N ,j = x j , 1 ≤ j ≤ N . (13) W e write the inte gral o f a function f of T ov er Γ N ( x ) as Z Γ N ( x ) f ( T ) d T ≡ N Y k =1 k Y j =1 Z ∞ −∞ dT k ,j f ( T ) N Y ℓ =1 δ ( T N ,ℓ − x ℓ ) . (14) 5 Then the in tegral represen tation of ψ ( N ) ~ λ ( x ) is giv en by ψ ( N ) ~ λ ( x ) = Z Γ N ( x ) e F ( N ) ~ λ ( T ) d T . (15) This multiv ar ia te function is a v ersion of Whittak er function (see [19] and references therein). As a sto c hastic ve rsion o f the Sc hr¨ odinger equation of the quan tum T o da lattice (obtained b y p erforming the Wick rotation in the Sc hr¨ odinger eq uation), w e consider the f ollo wing diffusion equation ∂ ∂ t u ( t, x ) = L N u ( t, x ) (16) with the infinitesimal generator o f the pro cess L N ≡ −H N = 1 2 ∆ − V N ( x ) , (17) where V N ( x ) = N − 1 X j =1 e − ( x j +1 − x j ) . (18) If we follow the metho d of separation of v ariables b y setting u ( t, x ) = T ( t )Ψ γ ( x ), (16) is decomp osed into the equations dT ( t ) dt = − γ T ( t ) and (10). Then we can conclude that for an y ~ λ ∈ C N , exp t 2 N X j =1 λ 2 j ! ψ ( N ) ~ λ ( x ) (19) solv es the diffusion equation (16). In the contex t of quan tum mec hanics, the function V N ( x ) given b y (18) pla ys, as a matter of course, a role of p oten tial energy . Then the quan tum system prefers the state x j +1 > x j to the state x j +1 < x j , 1 ≤ j ≤ N − 1, since the former has lo w er energy t han the la t t er. On the other hand, in the context o f sto c hastic calculus, − V N ( x ) term in the infinitesimal generator of the pro cess (17) acts as a kil ling term . W e consider N indep enden t one-dimensional standard BMs starting fro m 0, B j ( t ) , 1 ≤ j ≤ N , and fo r x ∈ R N set B x ( t ) = x + B ( t ), where eac h elemen t B x j j ( t ) = x j + B j ( t ) is a one-dimensional standard BM start ing fro m x j , 1 ≤ j ≤ N . Then the F eynman-Kac form ula (see, for instance, [14 ]) implies that the function Q N ( t, y | x ) = E  1 ( B y ( t ) = x ) exp  − Z t 0 V N ( B y ( s )) ds  (20) 6 solv es the diffusion equation (16) under the initial condition Q N (0 , y | x ) = δ ( x − y ) , (21) where E [ · ] denotes the exp ectation ov er all realizations of N -dimensional Bro wnian pa t hs, { B y ( s ) : 0 ≤ s ≤ t } , starting from y , a nd 1 ( ω ) is the indicator function of the ev en t ω ; 1 ( ω ) = 1 if ω is satisfied, 1 ( ω ) = 0 otherwise. The function Q N ( t, y | x ) is the transition probabilit y densit y of the pro cess (16) from a configuration x t o a configura t io n y in time in terv al t ≥ 0. In the F eynman-Kac form ula (20), w e consider a collection of all paths of BM in R N starting fr o m y to x . (Though the time direction is backw ard, it is irrelev an t in calculation, since BM is time-rev ersible.) The p oint of this form ula is the following. In order to giv e the transition probabilit y densit y Q N ( t, y | x ), w e ha v e to put a w eigh t w N = exp  − Z t 0 V N ( B y ( s )) ds  = exp ( − N − 1 X j =1 Z t 0 e − ( B y j +1 j +1 ( s ) − B y j j ( s )) ds ) (22) to eac h realization of path of the N -dimensional BM and tak e a summation o v er all re- alizations of paths. It is o bvious that w N tak es a real v alue in [0 , 1]. Then this sum- mation of w eigh ted pa t hs (a path in tegral) can b e iden tified with a statistical-ensem ble a v erage of Bro wnian paths, in whic h eac h pa th is included in the ensem ble with proba - bilit y w N and is deleted w ith pro babilit y 1 − w N . D eletion of an N -dimensional Bro wn- ian path is in terpreted as an eve n t that the N - dimensional BM is killed in the time in- terv al [0 , t ]. The w eight w N is then regarded as the probabilit y that the particle in R N surviv es up to time t . (See Coro llary 4.5 and explanation g iv en b elow it in Chapter 4 of [14] for the e quiv alence of the F eynman-Kac form ula to Brow nian motion with k illing of part icles.) Eq.(22) giv es the dependence of the surviv al probabilit y on the re alization of path { B y ( s ) , 0 ≤ s ≤ t } . If the N - tuples of Brownian paths a re “w ell-ordered” in the spatio-temp o r al plane, B y 1 1 ( s ) < B y 2 2 ( s ) < · · · < B y N N ( s ) , 0 ≤ s ≤ t , a nd moreov er B y j +1 j +1 ( s ) ≫ B y j j ( s ) , 1 ≤ j ≤ N − 1 , 0 ≤ s ≤ t , w N is large, while for a particle on the path { B y ( s ) , 0 ≤ s ≤ t } in whic h B y j +1 j +1 ( s ) < B y j j ( s ) , 1 ≤ j ≤ N for some s ∈ [0 , t ], w N is small. 7 If we in tro duce a par a meter ε > 0, then we can see that lim ε → 0 exp ( − N − 1 X j =1 Z t 0 e − ( B y j +1 j +1 ( s ) − B y j j ( s )) /ε ds ) = 1  B y 1 1 ( s ) , . . . , B y N N ( s ) do not collide during [0 , t ]  = 1  B y ( s ) ∈ W N , 0 ≤ ∀ s ≤ t  . (23) In this sense, the pro cess (16) with (17) and (18) is a n N -particle system of killing BMs, whic h can b e regarded as an extension of the absorbing BM in W N . In the next section, w e explain ho w to express the tr ansition probability densit y giv en b y the F eynman-Kac formula (20) as a sup erp osition of the T o da lattice eigenfunctions (19). Remark. If w e consider the presen t pro cess not as an N -dimensional BM in R N but as an N -particle system of o ne-dimensional BMs, (20) give s t he transition probabilit y densit y in the case that m utual killing of part icles do es not o ccur at all in time dura tion t , since x and y are b oth N -part icle configuratio ns, x , y ∈ R N . In order to discuss pro cesses, in whic h m utual killing of part icles actually o ccurs and total num b er of particles decreases in time, w e ha v e to sp ecify the w ay ho w to c ho o se pair of particles whic h are a nnihilated; e.g. the pair ( j, j + 1) atta ining min { B y k +1 k +1 ( t ) − B y k k ( t ) } is c hosen. Note that in the orig inal vicious BM, colliding pairs of particles are pair annihilated. In the presen t pa p er, ho w ev er, w e a r e in terested in the pro cess conditioned that all N part icles surviv e. I I I. TRANSITION PROBAB ILITY DENSITY AND ITS LONG-T ERM ASYMP- TOTICS The problem whic h is discussed here is ho w to determine the f unction g ~ λ ( y ) of ~ λ ∈ C N , y ∈ R N and a subset Σ of C N suc h that the in tegral o f (19) Z Σ exp t 2 N X j =1 λ 2 j ! ψ ( N ) ~ λ ( x ) g ~ λ ( y ) d ~ λ (24) is equal to Q N ( t, y | x ) giv en b y (20). This problem is solv ed b y applying the theory of the Skly anin measure [42] defined b y , s N ( ~ λ ) d ~ λ ≡ 1 (2 π i ) N N ! Y 1 ≤ j 0 a nd ν ≥ 0 , I ν ( x ) is a p ositiv e function whic h increases monotonically as x → ∞ , while K ν ( x ) is a p ositive function whic h decreases monotonically as x → ∞ . Since K ν ( z ) has the following integral represen tation K ν ( z ) = 1 2 Z ∞ 0 s ν − 1 exp  − z 2  s + 1 s  ds, (62) (57) is written as ψ (2) ( λ 1 ,λ 2 ) ( x 1 , x 2 ) = 2 e ( λ 1 + λ 2 )( x 1 + x 2 ) / 2 K λ 1 − λ 2 (2 e − ( x 2 − x 1 ) / 2 ) . (63) The infinitesimal generator of the pro cess (53) is then give n for N = 2 as G ( µ 1 ,µ 2 ) 2 = 1 2 ∆ + 2 X j =1 ∂ ∂ x j log ψ (2) λ 1 ,λ 2 ( x 1 , x 2 ) ∂ ∂ x j = 1 2  ∂ 2 ∂ x 2 1 + ∂ 2 ∂ x 2 2  + 1 2 ( µ 1 + µ 2 )  ∂ ∂ x 1 + ∂ ∂ x 2  + K ′ µ 1 − µ 2 (2 e − ( x 2 − x 1 ) / 2 ) K µ 1 − µ 2 (2 e − ( x 2 − x 1 ) / 2 ) e − ( x 2 − x 1 ) / 2  ∂ ∂ x 1 − ∂ ∂ x 2  , (64) where K ′ ν ( z ) ≡ dK ν ( z ) / dz . If w e change the v ariables ( x 1 , x 2 ) 7→ ( ξ , η ) b y ξ = ( x 1 + x 2 ) / 2, η = − ( x 1 − x 2 ) / 2 − log 2, (64) is decomp osed in to tw o parts, G ( µ 1 ,µ 2 ) 2 = ( G µ 1 + µ 2 0 + G µ 1 − µ 2 MY ) / 2, where G ν 0 = 1 2 d 2 dξ 2 + ν d dξ , (65) G µ MY = 1 2 d 2 dη 2 + d dη { log K µ ( e − η ) } d dη = 1 2 d 2 dη 2 − K ′ µ ( e − η ) K µ ( e − η ) e − η d dη . (66) The former is the infinitesimal g enerator of t he one-dimensional BM with a constan t drift ν = µ 1 + µ 2 and the la tter is tha t of the diffusion pro cess studied by Matsumoto and Y or with parameter µ = µ 1 − µ 2 [40, 41]. It implies that, in the N = 2 case of t he O’Connell pro cess with parameter ~ µ = ( µ 1 , µ 2 ) ∈ R 2 , the cen ter of mass ξ b ehav es a s (a time c hange t 7→ 2 t of ) a BM with a drif t µ 1 + µ 2 and the relativ e co or dina t e η b ehav es as (a time c hange t 7→ 2 t of ) the Matsumoto-Y or pro cess with parameter µ 1 − µ 2 . F or µ ∈ R , let L µ = 1 2 d 2 dx 2 − V ( x ) − µ d dx (67) with V ( x ) = 1 2 e − 2 x . (68) 16 It is the infinitesimal generator of t he one-dimensional drifted BM with a killing term − V ( x ). The transition probability density is giv en b y Q µ ( t, y | x ) = E  1 ( B µ,y ( t ) = x ) exp  − 1 2 Z t 0 e − 2 B µ,y ( s ) ds  = e − µ 2 t/ 2+ µ ( x − y ) i π 2 Z i ∞ − i ∞ e λ 2 t/ 2 K λ ( e − x ) K λ ( e − y ) λ sin( π λ ) d λ, (69) where B µ,y ( t ) = y + B ( t ) + µt with the one-dimensional standard BM, B ( t ), starting from 0; B (0) = 0 . The surviv al probability up to time T > 0 of this one-dimensional killing BM with drift µ is giv en by N µ ( T , x ) = Z ∞ −∞ Q µ ( T , y | x ) dy (70) for t he initial po sition x ∈ R . W e can prov e that, if µ > 0, it has the long-t erm asymptotics, lim T →∞ r π 2 T 3 / 2 e µ 2 T / 2 N µ ( T , x ) = e µx K 0 ( e − x ) Z ∞ −∞ K 0 ( e − y ) e − µy dy = 2 µ − 2 (Γ( µ/ 2)) 2 e µx K 0 ( e − y ) . (71) The pro o f of (7 1) is give n in App endix A . Then the transition probabilit y densit y for this one-dimensio nal kil lin g BM c onditione d to survive for ever is giv en in the limit µ → 0 , µ > 0 as P ( t, y | x ) = lim µ → 0 ,µ> 0 lim T →∞ N µ ( T − t, y ) N µ ( T , x ) Q µ ( t, y | x ) = K 0 ( e − y ) K 0 ( e − x ) i π 2 Z i ∞ − i ∞ e λ 2 t/ 2 K λ ( e − x ) K λ ( e − y ) λ sin( π λ ) d λ, (72 ) x, y ∈ R , t > 0. It is easy to see that (72) satisfies the diffusion equation ∂ ∂ t u ( t, x ) = G 0 MY u ( t, x ) , t ≥ 0 , (73) under the initial condition u (0 , x ) = δ ( x − y ). Matsumoto and Y or show ed that the sto chastic pro cess Z µ MY ( t ) = lo g  Z t 0 e 2 B µ ( s ) ds  − B µ ( t ) , t ≥ 0 (74) with B µ ( t ) ≡ B µ, 0 ( t ) = B ( t ) + µt is a diffusion pro cess, whose infinitesimal generator is giv en b y ( 6 6) for an y µ ∈ R . Here w e ha v e sho wn that, when µ = 0, the Matsumoto-Y or pro cess (74) can b e constructed as a one-dimensional killing BM conditioned to surviv e foreve r. See App endix B f o r more detail on the relation b etw een the Matsumoto-Y or pro cess and the N = 2 case o f the O’Connell pro cess. 17 VI. SPECIAL INITIAL CONDITIO NS In this section, first w e consider the one-dimensional diffusion pro cess with the infinites- imal generator (67) with (68). Let 0 < T < ∞ . Then the transition probability density of the pro cess conditioned to surviv e up to time T is giv en b y P µ T ( s, x ; t, y ) = N µ ( T − t, y ) N µ ( T − s, x ) Q µ ( t − s, y | x ) = N µ ( T − t, y ) Q µ ( t − s, y | x ) R ∞ −∞ Q µ ( T − s, z | x ) dz , (75) 0 ≤ s ≤ t ≤ T , x, y ∈ R , where Q µ and N µ are given b y (69) and (70), resp ectiv ely . The asymptotics of K λ ( e − x ) in x → −∞ is independent of λ [43] K λ ( e − x ) ≃ r π 2 e − x exp( − e − x ) as x → −∞ . (76) Then if w e define P µ T ( t, y | − ∞ ) ≡ lim x →−∞ P µ T (0 , x ; t, y ) , (77) it is giv en b y P µ T ( t, y | − ∞ ) = √ 2 π 2 µ − 2 (Γ( µ/ 2)) 2 T 3 / 2 e µ 2 ( T − t ) / 2 − π 2 /T θ e − y ( t ) e − µy N ( T − t, y ) , (78) where θ r ( t ) is given b y (A9) and (A1 0) in App endix A [44]. Since N µ (0 , y ) = 1 b y definition, w e obtain the distribution at time t = T , P µ T ( T , y | − ∞ ) = c µ ( T ) θ e − y ( T ) e − µy , y ∈ R , T > 0 (79) with c µ ( T ) = √ 2 π 2 2 − µ (Γ( µ/ 2)) − 2 T 3 / 2 e − π 2 /T . On the other hand, b y (71), if w e tak e the temp orally homogeneous limit T → ∞ in (78), then w e obtain the distribution P µ ( t, y | − ∞ ) ≡ lim T →∞ P µ T ( t, y | − ∞ ) = 2 e − µ 2 t/ 2 θ e − y ( t ) K 0 ( e − y ) , y ∈ R , t > 0 . (80) Next w e consider the N -par t icle system of the killing BMs with drift ~ µ ∈ W N conditioned to surviv e up to time T , 0 < T < ∞ . The transition probability densit y is giv en b y (42). Corresp onding to (76), the asymptotics of ψ ( N ) ~ λ ( x ) in x j → −∞ , 1 ≤ ∀ j ≤ N is independent of ~ λ (see Remark 8.1 in [19]) . T hen we o btain P ~ µ N ,T ( t, y | − ∞ ) ≡ lim x j →−∞ , 1 ≤ j ≤ N P ~ µ N ,T (0 , x ; t, y ) = e | ~ µ | 2 ( T − t ) / 2 J ~ µ ( N , T ) Θ N ( t, y ) exp( − ~ µ · y ) N ~ µ N ( T − t, y ) , (81) 18 where J ~ µ ( N , T ) = Z ( i R ) N e T P N j =1 λ 2 j / 2 s N ( ~ λ ) I ~ µ − ~ λ ( N ) d ~ λ (82) with I ~ µ − ~ λ ( N ) = Z R N ψ ( N ) − ~ λ ( z ) exp( − ~ µ · z ) d z , ~ λ ∈ ( i R ) N , ~ µ ∈ W N , (83) and [19] Θ N ( t, y ) = Z ( i R ) N e t P N j =1 λ 2 j / 2 ψ ( N ) − ~ λ ( y ) s N ( ~ λ ) d ~ λ. (84) A t time t = T , (8 1) giv es the distribution P ~ µ N ,T ( T , y | − ∞ ) = 1 J ~ µ ( N , T ) Θ N ( T , y ) exp( − ~ µ · y ) , y ∈ R N , T > 0 . (85) On the other hand, if w e tak e the limit T → ∞ , (81) giv es the distribution P ~ µ N ( t, y | − ∞ ) ≡ lim T →∞ P ~ µ N ,T ( t, y | − ∞ ) = e −| ~ µ | 2 t/ 2 Θ N ( t, y ) ψ ( N ) 0 ( y ) , y ∈ R N , t > 0 . (86) The three-dimensional Bessel pro cess is defined as the radial part of the three-dimensional BM and a bbreviated to BES(3). The ab o v e results will b e compared with the Imhof rela- tion b etw een BES(3) and the pro cess called a meander and its multiv a riate g eneralizations discusse d in [5, 10]. VI I. SUMMAR Y AND CONCLUDING REMARKS The vicious BM is obtained as a d iffusion scaling limit of Fisher’s vicious walk mo del [1, 5, 6]. It is an N -particle system of BMs in one dimension, whose p ositions a re a rranged in the order x 1 < x 2 < · · · < x N , suc h that if and only if t w o neighboring Brow nian par ticles collide with eac h other then they are pair annihilated, while they can enjo y free Brown ian motions if they are all lo cated separately from eac h other. In the presen t pap er w e ha v e considered a system of N Bro wnian particles with t he killing t erm − V N ( x ) = − N − 1 X j =1 e − ( x j +1 − x j ) . (87) That is, t he in teractions b et w een neighboring Bro wnian particles are lo ng-ranged and the risk to b e pair a nnihilat ed exists alw a ys, which is express ed by a ra pid decreasing function 19 (87) of the distance of the t w o particles x j +1 − x j . W e regard this system of mutually killing BMs as a generalized ve rsion of vicious BM, since the original vicious BM can b e identified with the system of BMs with the killing term obtained by − lim ε → 0 ,ε> 0 V N ( x /ε ). Though the original vicious BM has only contact interactions, if w e consider the system conditioned nev er t o collide with eac h other, then w e obtain a system of BMs with long- ranged in teractions; the SDE is giv en b y Eq.(8), in whic h b etw een a ny pair of part icles there acts a repulsiv e f o rce prop ortio nal to the in v erse of distance of the pair [8 ]. This N - particle pro cess is equiv alen t to the eigen v alue pro cess of an N × N Hermitian-matrix-v alued BM in tro duced b y Dyson in o r der to dynamically sim ulate the eigen v alue statistics of the Gaussian unitary ensem ble (G UE) of random matr ices (the D yson mo del) [15 –17]. As discussed in [45 , 46], the equiv alence b et w een the eigen v alue pro cess of Dyson and the noncolliding BM (the vicious BM conditioned nev er to collide) is the N -v ariate extension of the equiv alence b et w een BES(3) and the one-dimensional BM conditioned to sta y p ositiv e. In this sense, the Dyson mo del can b e regarded as a many -particle generalization of BES(3). Apart from the equiv alence b et w een the BES(3) and the conditional BM to stay p ositiv e, the f o llo wing equiv alence is es tablished. Let M ( t ) = max 0 ≤ s ≤ t B ( s ) , t ≥ 0, and de fine a pro cess Y ( t ) = 2 M ( t ) − B ( t ) , t ≥ 0. Then Y ( t ) is equiv alen t to BES(3), whic h is kno wn as Pitman’s ‘2 M − X ’ theorem [47 ] (see also [40, 41, 4 8]). As a multiv aria te extension of Pitman’s ‘2 M − X ’ theorem, another construction of the D yson mo del (the noncolliding BM) has b een rep o rted [25, 26, 2 8, 31, 32]. Matsumoto and Y or studied the sto chastic pro cess Z µ MY ( t ) , t ≥ 0 given b y (74). W e can see that lim ε → 0 ,ε> 0 εZ µ MY ( t/ε 2 ) = lim ε → 0 ,ε> 0  ε log  Z t 0 e 2 B µ ( s ) /ε ds  − B µ ( t ) − ε log ε 2  = 2 max 0 ≤ s ≤ t B µ ( s ) − B µ ( t ) , t ≥ 0 . (88) Then, when µ = 0, t his ε → 0 limit is equiv alen t to Y ( t ) and th us with the BES(3). In this sense, the Matsumoto-Y or pro cess is a generalization of the BES(3) [40, 41]. O’Connell [19] in tro duced an N -particle pro cess Z ~ µ ( t ) = ( Z ~ µ 1 ( t ) , . . . , Z ~ µ N ( t )) , t ≥ 0, ~ µ ∈ R N as a m ulti-dimensional generalization of the Matsumoto-Y or pro cess. Corresp o nding to the fact that the Matsumoto-Y or pro cess is a generalization of the BES(3), the O’Connell pro cess 20 is a generalization of the Dyson mo del. Actually , he sho w ed that lim ε → 0 ,ε> 0 ε Z 0 ( t/ε 2 ) , t ≥ 0 is equiv alent to the D yson mo del in the sense of an extension of Pitman’s ‘2 M − X ’ theorem [19]. W e p o in ted out that the BES(3), the original vicious BM, and the Dyson mo del (the noncolliding BM) can b e regarded as ultr a discr etization s [49] of the Matsumoto-Y or pro cess, the BMs with the killing term in the same form as t he quan tum T o da lattice p otential, and the O’Connell pro cess, resp ectiv ely . In the presen t pap er, w e discussed another construction of the O’Connell pro cess apart from the extension of Pitman’s ‘2 M − X ’ theorem. In the sp ecial case with ~ µ = 0, we hav e sho wn here that his pro cess is given as a generalized ve rsion of vicious BM conditioned to surviv e f o rev er. In order to demonstrate that the relatio n b et w een the presen t generalized vicious BM and the O’Connell pro cess is a mu ltiv aria te generalization of the relation b et w een a killing BM and the Matsumoto-Y or pro cess, w e sho w ed in Sec.V that the Matsumoto-Y or pro cess with µ = 0 is obtained as a killing BM conditioned to surviv e forever. W e w an t to emphasize that the presen t analysis is indeed based on the idea of O’Connell to discuss interacting diffusiv e particle s ystems using the exact solutions of the quan tum T o da lattices [19]. In Sec.I in the presen t pap er, w e listed up three fundamental prop erties of the noncol- liding BM. They are all inherited by the O’Connell pro cess in the extended form. (i) The Karlin-McGregor determinan tal expression (4) of q N ( t, y | x ), whic h is expanded b y the Sch ur functions (32), is generalized b y the in tegral for mula (31). (ii) The harmonic transform [13] from q N to p N (6) by the ha rmonic function h N giv en b y t he pro duct of differences of v ari- ables (the V a ndermonde determinan t ) (2) is now given b y the form ula (46) from Q N to P N . There h N is replaced b y an eigenfunction ψ ( N ) 0 of the infinitesimal generator L N (the Hamiltonian H N of the quan tum T o da lattice). (See [50 – 52] for harmonic transforms of one dimensional generalized diffusion pro cesses.) (iii) Theorem 2 in Sec.IV giv es the extended v ersion of the Kolmogo ro v equation of (7). There are a lot of future problems. In no ncolliding diffusion pro cesses, if w e study the situations starting from “the all zero state” and observ e particle distributions at a n arbitrary time 0 < t < ∞ in temp orally homogeneous pro cesses, and at the ending time t = T in temp orally inhomogeneous pro cesses defined only in an finite time-interv a l [0 , T ], w e hav e obtained the eigen v alue distributions of random matrices in a v ariety of ensem bles studied in random matrix theory [10]. In Se c.VI, w e demonstrated that “the all −∞ state” and 21 temp orally inhomogeneous v ersions of pro cesses will play imp ortant roles in the O’Connell pro cess. The noncolliding diffusion pro cess is determinan tal, in the sense that for a ny finite initial configuration all m ultitime correlation functions are give n by determinan ts asso ciate with an in tegral k ernel called the correlatio n k ernel [53, 54]. It will b e a c hallenging problem to clarify how matrix-structures ( i.e. symmetries o f sy stems) [16, 17] and solv a bility are inherited b y the family o f O’Connell pro cesses. Ac kno wledgmen ts The presen t author w ould lik e to thank H. T anem ura, T. Sasamoto, T. Imam ura for useful discussions on the prese n t w ork. A part of the presen t w ork w as done during the participation of the presen t author in ´ Ecole de Ph ysique des Houc hes on “ Vicious W a lk ers and Random Matrices” (May 16 -27, 2 0 11). The author t hanks G. Schehr, C. Donati- Martin, and S. P ´ ec h ´ e for well-organization o f the school and R. Chhaibi for useful discussion on the Matsumoto-Y or pro cess. This w ork is supp orted in part b y the Grant-in-Aid for Scien tific Researc h (C) (No.215 40397) of Japan So ciet y for the Promo t io n of Science. App endix A: Pro of of (71) Let J 0 ( z ) b e the Bessel function of the first kind of order 0, J 0 ( z ) = ∞ X k =0 ( − 1) k ( z / 2) 2 k ( k !) 2 , | z | < ∞ . (A1) Since the equalit y K λ ( x ) K λ ( y ) = π 2 sin( π λ ) Z ∞ log( y /x ) J 0 ( p 2 xy coth u − x 2 − y 2 ) sinh( uλ ) du (A2) holds for x > 0 , y > 0 , | Re λ | < 1 / 4 [43], (69) is written as Q µ ( t, y | x ) = e − µ 2 t/ 2+ µ ( x − y ) i 2 π 2 Z ∞ x − y du J 0 ( p 2 e − ( x + y ) cosh u − e − 2 x − e − 2 y ) × Z i ∞ − i ∞ dλ e λ 2 t/ 2 λ sinh( uλ ) = e − µ 2 t/ 2+ µ ( x − y ) √ 2 π t 3 / 2 Z ∞ x − y uJ 0 ( p 2 e − ( x + y ) cosh u − e − 2 x − e − 2 y ) e − u 2 / 2 t du, (A3) 22 where w e ha v e p erfo rmed the in tegral of λ o v er i R . T his gives lim t →∞ r π 2 t 3 / 2 e µ 2 t/ 2 Q µ ( t, y | x ) = e µ ( x − y ) 1 2 Z ∞ x − y uJ 0 ( p 2 e − ( x + y ) cosh u − e − 2 x − e − 2 y ) du. (A4) W e find that the λ → 0 limit of (A2 ) giv es the equalit y K 0 ( x ) K 0 ( y ) = 1 2 Z ∞ log( y /x ) uJ 0 ( p 2 e − ( x + y ) cosh u − e − 2 x − e − 2 y ) du, (A5) and then (A4) giv es lim t →∞ r π 2 t 3 / 2 e µ 2 t/ 2 Q µ ( t, y | x ) = e µ ( x − y ) K 0 ( e − x ) K 0 ( e − y ) . (A6) The asymptotics of K 0 ( e − y ) is kno wn as [43] K 0 ( e − y ) ≃    log(2 / e − y ) ∼ y as y → ∞ p π / (2 e − y ) exp( − e − y ) → 0 as y → −∞ . (A7) Then if µ > 0, R ∞ −∞ K 0 ( e − y ) e − µy dy < ∞ . Actually , for µ > 0, this integral is the Mellin transformation of K 0 ( z ) and w e obtain Z ∞ −∞ K 0 ( e − y ) e − µy dy = Z ∞ 0 K 0 ( z ) z µ − 1 dz = 2 µ − 2 (Γ( µ/ 2)) 2 . (A8) Therefore (71) is v alid. W e note that, for the f unction θ r ( t ) = i 2 π 2 Z i ∞ − i ∞ e λ 2 t/ 2 K λ ( r ) λ sin( π λ ) dλ, r > 0 , (A9) Y or gav e t he follow ing expression (see Eq.(6.b”) on page 43 of [44]), θ r ( t ) = r (2 π 3 t ) 1 / 2 e π 2 / 2 t Z ∞ 0 e − η 2 / 2 t e − r cosh η (sinh η ) sin  π η t  dη , r > 0 . (A10) Using this express ion, Matsumoto and Y o r rep orted the asymptotics (see Eq.(2.11) in [41]), lim t →∞ √ 2 π t 3 θ r ( t ) = K 0 ( r ) , r > 0 . (A11) Since the equalit y Q µ ( t, y | x ) = e − µ 2 t/ 2+ µ ( x − y ) Z ∞ 0 exp  − s 2 − 1 2 s ( e − 2 x + e − 2 y )  θ e − ( x + y ) /s ( t ) ds s (A12) is established, the limit (A6) can be concluded also from (A11). 23 App endix B: N = 2 case of t he O ’Connell pro cess By the equations (25), (28) and (63), we obtain Q 2 ( t, y | x ) = 1 2 π 3 Z i ∞ − i ∞ dλ 1 Z i ∞ − i ∞ dλ 2 e ( λ 2 1 + λ 2 2 ) t/ 2 e ( λ 1 + λ 2 ) { ( x 1 + x 2 ) − ( y 1 + y 2 ) } / 2 × K λ 1 − λ 2 (2 e − ( x 2 − x 1 ) / 2 ) K λ 1 − λ 2 (2 e − ( y 2 − y 1 ) / 2 )( λ 1 − λ 2 ) sin { π ( λ 1 − λ 2 ) } , (B1) x , y ∈ R 2 , t ≥ 0. If we c ha nge the integral v aria bles ( λ 1 , λ 2 ) 7→ ( λ, ν ) b y λ = λ 1 − λ 2 , ν = λ 1 + λ 2 , w e can calculate the integral with respect to ν . 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