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N dt dXj(t)= dBj(t)+ , j IN 1, 2,...,N , (1.1) Xj(t) Xk(t) ∈ ≡{ } k: k=j − Xk=16 where Bj(t), j = 1, 2,...,N are independent one-dimensional BMs. When the individual diffusion process is a squared Bessel process with index ν > 1, the noncolliding diffusion − process is called a noncolliding squared Bessel process, satisfying the SDE
N 4Zν(t) ds dZν(t)=2 Zν(t)dB (t)+2(ν + 1)dt + j , j I (1.2) j j j Zν(t) Zν(t) ∈ N k: k=j j − k q Xk=16 and if 1 <ν< 0 having a reflection wall at the origin. These processes dynamically sim- − ulate the eigenvalue statistics of the Gaussian random matrix ensembles studied in random matrix theory [17, 3]. Let M be the space of nonnegative integer-valued Radon measures on R. This space is a Polish space with the vague topology. The space can be regarded as a configuration space of unlabeled particles in R. A probability measure on the space M is called a determinan- tal point process (DPP) or Fermion point process, if its correlation functions are generally represented by determinants [30, 31]. In this paper we say that an M-valued process Ξ(t) is determinantal if the multitime correlation functions for any chosen series of times are repre- N N sented by determinants. It has been shown that for any initial configuration ξ = j=1 δxj , the unlabeled process ΞN (t) = N δ is a determinantal process for the Dyson model j=1 Xj (t) P in [10], and for the noncolliding Bessel process in [11]. Suppose that (Ξ(t), PξN ) is aP noncolliding diffusion process starting from the initial con- figuration ξN of a finite number of particles. Let ξ be the configuration of an infinite number of particles, and ξ[ L,L] denote the the restriction of ξ on the set [ L, L]. When the sequence − − of processes (Ξ(t), Pξ[−L,L] ), L N = 1, 2,... , converges to an M-valued process starting from the configuration ξ, we∈ denote{ the limit} process by (Ξ(t), Pξ), and say that process (Ξ(t), Pξ) is well-defined. Sufficient conditions were given for configuration ξ that limit pro- cess is well definied, for noncolliding BM with bulk scaling [10], noncolliding BM with soft edge scaling [9], and noncolliding Bessel processes with hard-edge scaling [11]. We denote Pξ Pξ Pξ the assocoated limit processes by (Ξ(t), sin), (Ξ(t), Ai) and (Ξ(t), ν), respectively.
Let µsin, µAi and µJν be the DPPs with the sine kernel (2.1), the Airy kernel (2.2) and the Bessel kernel (2.3), respectively. They are the probability measures obtained in the bulk scaling limit and soft-edge scaling limit of the eigenvalue distribution (3.10) in the Gaussian unitary ensemble (GUE), and in the hard-edge scaling limit of the eigenvalue distribution (3.15) in the chiral Gaussian unitary ensemble (chGUE), respectively. It was shown in [12] Pξ Pξ Pξ that processes (Ξ(t), sin), (Ξ(t), Ai) and (Ξ(t), ν) have DPPs µsin, µAi and µJν as reversible measures, and the reversible processes coincide with determinantal processes (Ξ(t), Psin) with the extended sine kernel (2.9), (Ξ(t), PAi) with the extended Airy kernel (2.10), and (Ξ(t), Pν) with the extended Bessel kernel (2.11), respectively. The main purpose of this paper is
2 to prove the strong Markov property of these infinite-dimensional determinantal processes (Ξ(t), P⋆), ⋆ sin, Ai Jν ; ν > 1 (Theorem 2.1). For each∈{⋆ sin}∪{, Ai J ;−ν} > 1 , the diffusion process (Ξ(t), Pξ) associated ∈ { }∪{ ν − } ⋆ with DPP µ⋆ was constructed by a Dirichlet form technique presented by the first author [23, 24], and associated infinite-dimensional stochastic differential equation (ISDE) was de- Pξ ξ rived [22, 26, 5]. The relation between processes (Ξ(t), ⋆) and (Ξ(t), P⋆) was first discussed in [8]; it was later proven that both of these are extensions of the same pre-Dirichlet form on the set of polynomial functions [25]. However, the coincidence of these two processes has long been an open problem. Using Theorem 2.1, it can be proved affirmatively (Theo- rem 2.2). The coincidence of processes from completely different approaches of construction enable us to examine them from various points of view. From the algebraic construction by virtue of their determinantal structure of their space-time correlation functions, we ob- tain quantitative information of the processes such as the moment generating functions of multitime distributions (2.8) given by the Fredholm determinant of space-time correlation functions [9, 10, 11]; while from the analytic construction through Dirichlet form theory, we deduce many qualitative properties of sample paths of the labeled diffusion by means of the ISDE representation, which makes us possible to apply the Ito calculus to the processes [20, 22, 26]. The proof is based on the following two facts: Pξ ξ (i) Determinantal process (Ξ(t), ⋆) solves the same ISDE as process (Ξ(t), P⋆). (ii) The solutions of the ISDE are unique. Fact (ii) was shown in [27]. Fact (i) is derived by the arguments in [22], in which the consistent families of diffusion processes (Xk(t),H(t)), k N describing the joint processes of tagged ∈ particles and unlabeled infinite particles are introduced. To construct the processes, the Pξ strong Markov property of the unlabeled process (Ξ(t), ⋆) is crucial [21]. Pξ To prove the strong Markov property of the determinantal process (Ξ(t), ⋆) we introduce the subsets X⋆ in (2.5)–(2.7), and equip them with the topology defined by the inductive limit. One of the key points for the strong Markov property is to prove the Feller-like Pξ X property of the process (Ξ(t), ⋆) restricted on ⋆. For showing the Feller property of the ξn ξ process it is necessary to have if ξn ξ in X⋆ with the topology, P P weakly on C([0, ), X ). In stead of the property,→ we prove that Pξ (Ξ C([0, ], X →)) = 1 for µ -a.s. ∞ ⋆ ⋆ ∈ ∞ ⋆ ⋆ ξ in Proposition 3.7, and combine it with the fact that if ξ ξ in X with the topology, n → ⋆ Pξn Pξ weakly on C([0, ], X), which was already proved in [9, 10, 11], and derive the → ∞ strong Markov property of the revrsible process (Ξ(t), P⋆). N N N Let µsin, µAi and µJν be probability measures on the configuration spaces of finite numbers of particles defined by (3.11), (3.13) and (3.16), respectively. For each ⋆ sin, Ai Jν ; ν > N ∈{ }∪{ 1 , µ⋆ is obtained by the limit of µ⋆ , N . It is interesting to study the relation between − } N →∞ the Dirichlet forms related to µ⋆ and µ with the same square field defined in (2.12). We ξN N denote by (Ξ(t), P⋆ ) the Diffusion process associated the Dirichlet form related to µ⋆ and starting from any configuration of N particles. As one example of byproducts of Theorem ξl 2.2, we obtain that (Ξ(t), P N ) converges to (Ξ(t), Pξ), N , where ξl is the configuration ⋆ ⋆ →∞ N given in (2.13) (Corollary 2.3).
3 The paper is organized as follows. In Section 2 preliminaries and main results are given. In section 3 we give the proof of the main theorems.
2 Preliminaries and main results
Let M = M(R) be the space of nonnegative integer-valued Radon measures on R, which is a Polish space with the vague topology. We say ξn, n N converges to ξ vaguely if R ∈ R limn R ϕ(x)ξn(dx) = R ϕ(x)ξ(dx) for any ϕ C0( ), where C0( ) is the set of all →∞ continuous real-valued functions with compact supports.∈ A subset N of M is relatively R R R compact if and only if supξ N ξ(K) < , for any compact set K in . Each element ξ of M ∈ ∞ can be represented as ξ( ) = j Λ δxj ( ) with an index set Λ and a sequence of points · ∈ · x = (xj)j Λ in R satisfying ξ(K) = ♯ xj : xj K < for any compact subset K R. ∈ P { ∈ } ∞ ⊂ We introduce the subspace M0 of M defined as M0 = ξ M : ξ( x ) 1, x R . Any element ξ M is identical to its support, and can{ be regarded∈ as{ } a countable≤ subs∈ et} ∈ 0 xj j Λ. For an element ξ M M0, there is a point x supp ξ such that ξ( x ) 2. Such { } ∈ ∈ \ ∈ { } ≥ a point is called a multiple point. We call an element ξ of M an unlabeled configuration, and M a sequence of points x = (xj)j Λ a labeled configuration. For ξ( ) = j Λ δxj ( ) , we ∈ · ∈ · ∈ introduce the following operations P (restriction) ξ = ξ A = δ ( ), for A R, A ∩ x · ⊂ x A X∈ (shift) τ ξ( )= δ ( ), for u R. u · xj +u · ∈ j Λ X∈ We also introduce the configuration space M+ = ξ M : ξ(( , 0)) = 0 . A probability measure on the space M is called{ ∈ a DPP or−∞Fermion point} process, if its correlation functions are generally represented by determinants [30, 31]. In this paper we consider the DPPs with the following three continuous kernels on R R. × (i) Sine kernel Ksin(x, y):
sin(x y) − if x = y, π(x y) 6 1 √ 1k(x y) Ksin(x, y)= dk e − − = − (2.1) 2π k 1 Z| |≤ 1 if x = y. π (ii) Airy kernel KAi(x, y):
Ai(x)Ai′(y) Ai′(x)Ai(y) − if x = y, x y 6 KAi(x, y)= − (2.2) 2 2 (Ai′(x)) x(Ai(x)) if x = y, − 4 1 √ 1(xk+k3/3) R where Ai(x)= 2π R dke − , x , is the Airy function and Ai′(x)= dAi(x)/dx. (iii) Bessel kernel K (x, y): ∈ R Jν
Jν(2√x)√yJ ′ (2√y) √xJ ′ (2√x)Jν(2√y) ν − ν if x = y, x y 6 KJν (x, y)= − (2.3) 2 Jν(2√x) Jν+1(2√x)Jν 1(2√x) if x = y, − − where J ( ) is the Bessel function with index ν > 1 defined as ν · − ∞ ( 1)n z 2n+ν J (z)= − , z C ν Γ(n + 1)Γ(n +1+ ν) 2 ∈ n=0 X ∞ u x 1 with the gamma function Γ(x)= 0 e− u − du and Jν′ (x)= dJν(x)/dx. We denote by µsin,
µAi, µJν , DPPs with the sine, Airy and Bessel kernels, respectively. We note that DPPs µsin, R N N N µAi and µjν are the limit distributions of DPPs µsin, µAi, and µJν defined, respectively, in (3.11), (3.13), (3.16), which are related to the eigenvalue distributions of random matrices [17, 1]. Let ρ(dx) = ρ(x)dx be a Radon measure on R with density ρ(x) such that ρ(R) = ε,κ ρ(x)dx 0 N . For ε,κ > 0 and m0, L0 N, we denote by X (ρ) the set of R ∈{ } ∪ ∪ {∞} ∈ L0,m0 configurations ξ M satisfying ξ(R)= ρ(R), R ∈ ρ([0, L]) ξ([0, L]) Lε, ρ([ L, 0]) ξ([ L, 0]) Lε, L L , | − | ≤ | − − − | ≤ ≥ 0 and κ κ max ξ [g (k),g (k + 1)] m0, k Z ≤ ∈ where gκ(x) = sign(x) x κ, x R. (2.4) | | ∈ Xε,κ The set L0,m0 (ρ) is then relatively compact with the vague topology. We introduce the following three configurations associated with DPPs µ , ⋆ sin, Ai J ; ν > 1 : ⋆ ∈{ }∪{ ν − } X Xε,κ sin = L0,m0 (ρsin), (2.5) L0,m0 N ε (0,1) [∈ κ ∈(0[,κ ) ∈ sin X Xε,κ Ai = L0,m0 (ρAi), (2.6) L0,m0 N ε (0,1) [∈ κ ∈(0[,κ ) ∈ Ai + ε,κ XJ = M X (ρJ ), (2.7) ν ∩ L0,m0 ν L0,m0 N ε (0,1) [∈ κ ∈(0[,κ ) ∈ Jν where κsin = 1, κAi =2/3, κJν = 2, and ρ⋆ is the first order correlation function of DPPs µ⋆ for each ⋆ sin, Ai J ; ν > 1 . It is readily verified that µ (X ) = 1, for ⋆ = sin, ∈ { }∪{ ν − } ⋆ ⋆ 5 Ai or Jν. We equip space X⋆ with the topology defined by the inductive limit. Note that this is a locally compact space with the topology. We denote by C([0, ), X⋆) the set of X -valued continuous functions with the topology. Then Ξ( ) C([0, ∞), X ) implies that ⋆ · ∈ ∞ ⋆ Ξ( ) is continuous with the vague topology, and for any T > 0 there exist M0, L0 N, · Xε,κ ∈ ε (0, 1) and κ (0, κ ) such that Ξ(t) L ,m (ρ ) for t [0, T ]. ∈ ∈ ∗ ∈ 0 0 ∗ ∈ In this paper we say that an M-valued process Ξ(t) is determinantal, if the multitime correlation functions for any chosen series of times are represented by determinants. In other words, a determinantal process is an M-valued process such that, for any integer M M N, f = (f1, f2,...,fM ) C0(R) , and sequence of times t = (t1, t2,...,tM ) with 0 <∈ t < < t < , the moment∈ generating function of multitime distribution, Ψt[f] 1 ··· M ∞ ≡ E M exp m=1 R fm(x)Ξ(tm, dx) , is given by a Fredholm determinant
h nP R t oi Ψ [f]= Det δstδ(x y)+ K(s, x; t, y)χt(y) , (2.8) 2 (s,t) t1,t2,...,tM − ∈{ } (x,y) R2 ∈ h i where χ = efm 1, 1 m M, and K is a locally integrable function called a (space-time) tm − ≤ ≤ correlation kernel [8, 9, 10, 11]. In this paper we study the determinantal processes with the following three correlation kernels: (i)Extended sine kernel K (s, x; t, y),s,t R+ x R : x 0 ,x,y R: sin ∈ ≡{ ∈ ≥ } ∈ 1 1 u2(t s)/2 du e − cos u(y x) if s < t, π { − } K Z0 sin(s, x; t, y)= Ksin(x, y) if s = t, (2.9) 1 ∞ u2(t s)/2 du e − cos u(y x) if s > t. −π 1 { − } Z + (ii) Extended Airy kernel KAi(s, x; t, y),s,t R ,x,y R: ∈ ∈ ∞ u(t s)/2 du e− − Ai(u + x)Ai(u + y) if s < t, Z0 KAi(s, x; t, y)= KAi(x, y) if s = t, (2.10) 0 u(t s)/2 du e− − Ai(u + x)Ai(u + y) if s > t. − Z−∞ + + (iii) Extended Bessel kernel KJ (s, x; t, y),s,t R ,x,y R : ν ∈ ∈ 1 2u(s t) du e− − Jν(2√ux)Jν(2√uy) if s < t, 0 K (s, x; t, y)= Z (2.11) Jν KJν (x, y) if s = t, ∞ 2u(s t) du e− − Jν(2√ux)Jν(2√uy) if s > t. − 1 Z K We denote the determinantal process with the extended sine kernel ⋆(s, x; t, y) by (Ξ(t), P⋆), ⋆ = sin, Ai,Jν. The determinantal process (Ξ(t), P⋆) is reversible with the DPP µ⋆, for ⋆ = sin, Ai and Jν. The main theorem of this present paper is as follows.
6 Theorem 2.1 Let ⋆ sin, Ai Jν ; ν > 1 . The determinantal process (Ξ(t), P⋆) is a reversible diffusion process.∈{ }∪{ − }
Diffusion processes with reversible probability measures µ⋆,⋆ sin, Ai Jν ; ν > 1 were constructed by means of the Dirichlet form technique by the∈{ first author}∪{ [23, 24]− and} associated ISDEs were derived in [22, 26, 5]. The second theorem of this paper is the coincidence of the process constructed by the Dirichlet form technique and the determinantal process (Ξ(t), P ), ⋆ sin, Ai J ; ν > 1 . To state the theorem we prepare some ⋆ ∈ { }∪{ ν − } notation and review the related results. A function f on M is called local if f(ξ)= f(ξK) for some compact set K. For a local function f with f(ξ)= f(ξK) we introduce the functions fˇ on Rk, k N 0 N defined as fˇ = f( ), where is the null configuration, and for k ∈ 0 ≡{ } ∪ 0 ∅ ∅ k N, as ∈ k fˇ (x )= f δ for x Kk. k k xj k ∈ j=1 ! X ˇ k k We extend the domain of fk(xk) to R K by the consistency arising from f(ξ) = f(ξK). Hence fˇ , k N , satisfy the consistency\ relation k ∈ 0 fˇ (x ,y)= fˇ (x ), x Rk, y/ K. k+1 k k k k ∈ ∈ ˇ A local function f is called smooth if the fk are smooth for k N0. We denote by the ∞ set of all local smooth functions on M. ∈ D k Rk N Let xk =(xi)i=1 , k 0, and f,g ; then ∈ ∈ ∈D∞ 1 k ∂fˇ (x ) ∂gˇ (x ) Da(f,g)(x )= a(x ) k k k k , (2.12) k 2 i ∂x ∂x i=1 i i X where a is a smooth function on R. For given f,g , the right-hand side is a permutation ∞ invariant function, and the square field D(f,g)∈D can be regarded as a local function with M M 2 M variable ξ = i N δxi . For probability µ on , we denote by L ( ,µ) the space of ∈ ∈ square integrable functions on M with the inner product , µ and the norm L2(M,µ). We P a µ 2 M h· ·i k·k consider the bilinear form ( µ, ) on L ( ,µ) defined as E D∞ a Da µ(f,g)= (f,g)dµ, E M µ Z 2 = f : f 1 < , D∞ { ∈D∞ k k ∞} 2 a 2 where f (f, f)+ f 2 . We consider three cases (µ, a(x)) = (µ , 1), (µ , 1) k k1 ≡ Eµ k kL (M,µ) sin Ai and (µJν , 4x). For each case it is proved in [23, 24] that the closure ( ⋆, ⋆) of the bilinear a µ⋆ E D ξ form ( µ⋆ , ) is a quasi-Dirichlet form, associated with the diffusion process (Ξ(t), P⋆), ⋆ sinE , AiD∞ J ; ν > 1 . One can naturally lift each unlabeled path Ξ to the labeled ∈ { }∪{ ν − } path X by a labeled map l =(l)i N (see for example [21, Theorem 2.4]). It was also proved ∈ N that a labeled process X =(Xj)j with Ξ(t)= j N δXj (t) solves the ISDE ∈ ∈ P∞ dt dXj(t)= dBj(t) + lim , (sin) r Xj(t) Xk(t) →∞ k: k=j − XX(t6)
lj 1(ξ) Llj(ξ) , j =1, 2,...,ξ(R), | − |≤| | and set N l ξ = δl for ξ M. (2.13) N j (ξ) ∈ j=1 X N N N ξ Let X = (Xj)j N and X = (Xj )j=1 be the labeled processes associated with (Ξ(t), P⋆) ∈ and (Ξ(t), PξN ), respectively. Note that X(0) = l(ξ) x and XN (0) = (l (ξ))N xN . We ⋆ ≡ j j=1 ≡ have then the following as a corollary of Theorem 2.2.
8 Corollary 2.3 Let ⋆ sin, Ai J ; ν > 1 . (i) For µ a.s. ξ, ∈{ }∪{ ν − } ⋆ ξl (Ξ(t), P N ) (Ξ(t), Pξ), N , ⋆ → ⋆ →∞ weakly on the path space C([0, ), M). 1 ∞ (ii) For µ l− a.s. x, and m N ⋆ ◦ ∈ (XN (t),XN (t),...,XN (t)) (X (t),X (t),...,X (t)), N , 1 2 m → 1 2 m →∞ weakly on the path space C([0, ), Rm). ∞ Suppose that µ and µN (N N) are probability measures on M, and (Ξ(t),P ) and (Ξ(t),P N ) are diffusion processes∈ associated with the Dirichlet spaces given by the closures a µ 2 M a µN 2 M N of ( µ, , L ( ,µ)) and ( µ , , L ( ,µ )), respectively. Let us consider the problem E D∞ E N D∞ on the weak convergence of stationary processes. That is, µ µ, N (Ξ(t),P N ) (Ξ(t),P ), N . N → →∞ ⇒ → →∞ If the measures µN and µ are singular each other, then such a convergence is not covered by a general theorem of convergence of diffusions associated with Dirichlet forms. We remark that Corollary 2.3 (i) gives examples of such a convergence even if the measures µN and µ are singular each other. Recently, Kawamoto-Osada [14] also showed Corollary 2.3 (ii) using a different method.
3 Proof of theorems
3.1 Some properties of determinantal point processes Let µ be a probability measure on M with correlation functions ρ (x ), x Rm, m N. m m m ∈ ∈ Then, for f C (R) and θ R ∈ 0 ∈ m ∞ 1 θ RRd f(x)ξ(dx) θf(xj ) Ψ(f, θ) µ(dξ)e = dxm e 1 ρm xm , ≡ M m! Rm − m=0 j=1 Z X Z Y where dx = m dx . Let be a symmetric linear operator with kernel K. In this section m j=1 j K we assume that operator satisfies Condition A in [30]. Probability measure µ is called a DPP with correlationQ kernelK K, if its correlation functions are given by
ρm(xm) = det K(xj, xk) . 1 j,k m ≤ ≤ h i R We often write ρ for ρ1, and ρ(A) for A ρ(x)dx, A ( ). The following lemmas are [12, Lemma 3.1] and a modification of [12, lemma 3.3]. ∈ B R Lemma 3.1 Let µ be a DPP. For any bounded closed interval D of R, we then have
2k µ(dη) η(D) ρ(x)dx (3ρ(D))k, k N. M − D ≤ ∈ Z Z
9 Lemma 3.2 Let µ be a DPP and (Ξ(t),P ) be a stationary process with stationary measure µ. If k ℓρ([gκ(k),gκ(k + 1)])m < , (3.1) | | ∞ k Z X∈ for some ℓ, m N, and κ> 0, then for P -a.s. Ξ there exists m = m (Ξ) N such that ∈ 0 0 ∈ j Ξ(t, [gκ(k),gκ(k + 1)]) m , t = , j =1, 2,..., k ℓ,k Z, (3.2) ≤ 0 k ℓ | | ∈ | | where gκ is the function in (2.4). In particular j lim P Ξ(t, [gκ(k),gκ(k + 1)]) m, t = , j =1, 2,..., k ℓ,k Z =1. (3.3) m ≤ k ℓ | | ∈ →∞ | | Proof. Since a DPP has the repulsive property, from condition (3.1) we derive
ℓ k ρm(xm)dxm < , | | κ κ m ∞ k Z [g (k),g (k+1)] X∈ Z which implies
k ℓµ ξ(gκ(k),gκ(k + 1)) > m | | k Z X∈ j = P Ξ(t, [gκ(k),gκ(k + 1)]) > m, for some t ; j =1, 2,..., k ℓ < . ∈ k ℓ | | ∞ k Z X∈ | | By Borel-Cantelli’s lemma we obtain (3.1), and then (3.2).
3.2 Non-equilibrium dynamics In this subsection we review the results in [9, 10, 11] on non-equilibrium dynamics of non- colliding diffusion processes. For ξN M with ξN (R) = N N and p N N 0 , we ∈ ∈ ∈ 0 ≡ ∪{ } consider the product w ξN ( x ) Π (ξN ,w)= G ,p { } , w C, p x ∈ x supp ξN ∈ Y where 1 u, if p =0 − G(u,p)= u2 up (1 u) exp u + + + , if p N. − 2 ··· p ∈ The functions G(u,p) are called the Weierstrass primary factors [15]. We then set ξN ( x ) N N c w z { } Φp(ξ ,z,w) Πp(τ zξ 0 ,w z)= G − ,p , w,z C, ≡ − ∩{ } − x z ∈ x supp ξN z c ∈ Y ∩{ } − 10 C where τzξ( ) j Λ δxj +z( ) for z and ξ( )= j Λ δxj ( ). u · ≡ ∈ · ∈ Rn · M ∈ · u n Let be the unlabeled map from n∞=0 to defined as (x1,...,xn)= j=1 δxj . We P u ∪ P consider the process (X(t)), t [0, ), for the solution X(t)=(Xj(t))j IN of (1.1). We N ∈ ∞ ∈ Pξ P N donote by sin the distribution of the process starting from any fixed configuration ξ with PξN a finite number of particles. It was proved in Proposition 2.1 of [10] that process (Ξ(t), sin) is determinantal with the correlation kernel KξN given by
N KξN 1 Π0(τ zξ , iu z) sin (s, x; t, y) = du dzpsin(s, x z) − − psin( t, iu y) 2πi R N | iu z − | Z ICiu(ξ ) − 1(s > t)p (s t, x y), (3.4) − sin − | N where Cw(ξ ) denotes a closed contour on the complex plane C encircling the points in N supp ξ on the real line R once in the positive direction but not point w, and psin(t, x y) is the generalized heat kernel: | 1 (x y)2 psin(t, x y)= exp − 1(t = 0)+ δ(y x)1(t = 0), t R, x,y C. | 2π t − 2t 6 − ∈ ∈ | | n o In case ξN is a configurationp without any multiple points, i.e. ξN M , (3.4) is rewritten as ∈ 0 KξN N N sin (s, x; t, y) = ξ (dx′) dupsin(s, x x′)Φ0(ξ , x′, iu)psin( t, iu y) R R | − | Z Z 1(s > t)p (s t, x y). − sin − |
It was proved in [10, Theorem 2.4] with [13, Theorem 1.4] that if ξ Xsin, process ξl ∈ P N Kξ (Ξ(t), sin) converges to the determinantal process with correlation kernel as N , ξ → ∞ weakly on C([0, ), M). In particular, when ξ X M , K is given by ∞ ∈ sin ∩ 0 sin Kξ sin(s, x; t, y) = ξ(dx′) dupsin(s, x x′)Φ0(ξ, x′, iu)psin( t, iu y) R R | − | Z Z 1(s > t)p (s t, x y). − sin − |
Letρ ˆN , N N be a sequence of nonnegative functions on R defined as ∈
1( 4N 2/3,0](x) x ρˆN (x)= − x 1+ . π − 4N 2/3 r N ρˆN (x) 1/3 Then, R dx ρˆ (x)= N, R dx x = N , and − R ρˆNR(x) ρˆ(x)=(√ x/π)1(x< 0), N . ր − →∞
Consider the process Y(t)=(Yj(t))j I given by ∈ N t2 ρˆN (x)dx Yj(t)= Xj(t)+ + t , j IN , 4 R x ∈ Z 11 with the solution X(t)=(Xj(t))j I of (1.1). In other words, Y(t) satisfies the following ∈ N SDE:
N t 1/3 dt dYj(t)= dBj(t)+ N dt + , j IN . (3.5) 2 − Yj(t) Yk(t) ∈ k: k=j − Xk=16
PξN u We denote by Ai the distribution of the process (Y(t)), t [0, ) starting from any fixed configuration ξN with a finite number of particles. It was proved∈ ∞ in Proposition 2.4 in [9] PξN KξN that process (Ξ(t), Ai ) is determinantal with correlation kernel ρˆN given by
ξN 1 Π0(τ zξ,iu z) K − ρˆN (s, x; t, y) = du dz q(0,s,x z) − 2πi R N − iu z Z ICiu(ξ ) − ρˆN (v)dv exp (iu z) q(t, 0, iu y) × − R v − Z 1(s > t)q(t, s, x y), − − where q(s,t,y x),s,t R,s = t, x, y C, is given by − ∈ 6 ∈ t2 s2 q(s,t,y x)= p t s, y x . − sin − − 4 − − 4 Note that q(s,t,y x), 0 s < t, x, y R is the transition density function of process B(t)+ t2/4, where−B(t), t ≤[0, ) is the one-dimensional∈ standard BM. Let M (ξ) be the function∈ defined∞ as A ρˆ(x)dx ξ(dx) M (ξ) = lim − . A L 0< x c ΦAi(ξ,w) exp wM (ξ) Π1(ξ 0 ,w), w C, ≡ " A # ∩{ } ∈ ΦAi(ξ,z,w) ΦAi(τ zξ,w z), w,z C. ≡ − − ∈ We note that ΦAi(ξ,z,w) exists finitely and Φ (ξ,z,w) 0, if ξ XAi. A ξl 6≡ ∈ X P N If ξ Ai, sequence of the processes (Ξ(t), Ai ) converges to the determinantal process Pξ∈ Kξ M (Ξ(t), Ai) with correlation kernel Ai as N , weakly on C([0, ), ). In particular, when ξ X M , Kξ is given by → ∞ ∞ ∈ Ai ∩ 0 Ai Kξ Ai(s, x; t, y)= ξ(dx′) du q(0,s,x x′)ΦAi(ξ, x′, iu)q(t, 0, iu y) R R − − Z Z 1(s > t)q(t, s, x y). − − 12 This result was stated in [9, Section 2.4], in which the weak convergence is in the sense of finite dimensional distributions. The convergence on C([0, ), M) is derived using the same argument as [13, Theorem 1.4]. ∞ We consider a one-parameter family of M+-valued processes with parameter ν > 1, u ν ν ν − (Z (t)), t [0, ), for the solution Z (t)=(Zj (t))j IN of SDE (1.2). For a given configu- ∈ ∞ ∈ N N M+ Pξ ration ξ with a finite number of particles, we denote by Jν the distribution of the ∈ N N Pξ process starting from ξ . In [11, Theorem 2.1] it was proved that (Ξ(t), Jν ) is determinantal with correlation kernel 0 N ξN 1 (ν) Π0(τ zξ ,u z) (ν) K − Jν (s, x; t, y) = du dzp (s, x z) − p ( t, u y) N 2πi Cu(ξ ) | u z − | Z−∞ I − 1(s > t)p(ν)(s t, x y), (3.6) − − | where for t R and x, y C ∈ ∈ 1 y ν/2 x + y √xy p(ν)(t, y x) = exp I 1(t =0, x = 0) | 2 t x − 2t ν t 6 6 | | | | yν y + exp 1(t =0, x = 0)+ δ(y x)1(t = 0). (3.7) (2 t )ν+1Γ(ν + 1) −2t 6 − | | (ν) Note that for t 0 and x, y R+, p (t, y x) is the transition density function of a 2(ν + ≥ ∈ | ξl X P N 1)-dimensional squared Bessel process. If ξ Jν , process (Ξ(t), Jν ) converges to the ∈Kξ M determinantal process with correlation kernel ν as N weakly on C([0, ), )). In particular, when ξ X M , Kξ is given by → ∞ ∞ ∈ Jν ∩ 0 Jν 0 ξ ∞ (ν) (ν) K ′ ′ ′ Jν (s, x; t, y) = ξ(dx ) dup (s, x x )Φ0(ξ, x ,u)p ( t, u y) 0 | − | Z Z−∞ 1(s > t)p(ν)(s t, x y). − − | The result was proved in [11, Theorem 2.2] in which the weak convergence is in the sense of finite dimensional distributions. The weak convergence on C([0, ), M) can be proved from ξl ∞ P N the tightness of J N N, which is derived by the same procedure to show [13, Theorem ν ∈ 1.4] using determinatal{ } martingales representation in [7, Theorem 1.2] instead of complex Brownian motion representation in [13, Theorem 1.1]. The following proposition is a combination of the results in [9, 10, 11, 12] and the tightness Pξn of ⋆ n N, ⋆ sin, Ai Jν ; ν > 1 , which is derived by the same argument as that { l } ∈ ∈ { }∪{ − } ξN of P⋆ N N. { } ∈ Proposition 3.3 Let ⋆ sin, Ai J ; ν > 1 . ∈{ }∪{ ν − } (i) Suppose that ξ,ξ X , n N. If ξ converges to ξ in X , then n ∈ ⋆ ∈ n ⋆ (Ξ(t), Pξn ) (Ξ(t), Pξ ), n (3.8) ⋆ → ⋆ →∞ weakly on C([0, ), M). ∞ (ii) Determinantal process (Ξ(t), P⋆) with the correlation kernel K⋆ is identical in distribution Pµ⋆ Pν Pξ M to process (Ξ(t), ⋆ ), where ⋆ = M ⋆ ν(dξ) for probability measure ν on . R 13 3.3 Path property of noncolliding processes Let X(t)=(X1(t),X2(t),...,XN (t)) be the Dyson model defined by (1.1). We introduce the following version of the Dyson model γN t X˜j(t)= e− Xj(τN (t)), 1 2tγN ˜ ˜ where γN = 2N and τN (t)=(e 1)/2γN . Process X(t)=(Xj(t))j=1,...,N then solves the SDE − X˜ (t)dt N dt dX˜ (t)= dB˜ (t) j + , j I , (3.9) j j − 2N ˜ ˜ ∈ N k: k=j Xj(t) Xk(t) Xk=16 − where the B˜j(t)’s are independent one-dimensional standard BMs. The eigenvalue distribu- tion in GUE, 1 1 N mN (dx )= x x 2 exp x2 dx (3.10) GUE N Z | i − j| −2N k N 1 i N N 1 µ m u− (3.11) sin ≡ GUE ◦ N as the reversible measure. Measure µsin is the DPP with the correlation kernel N 1 1 − x y KN (x, y)= ϕk ϕk , √2N √2N √2N Xk=0 k where hk = √π2 k! and 1 x2/2 ϕk(x)= e− Hk(x), √hk with Hermite polynomials Hk,k N0. ˜ ∈ ˜ We also introduce process Y(t)=(Yj(t))j IN obtained from the Dyson model in (3.9) by 1/3 2/3 ∈ 2/3 the transformation given by N − X˜ (N t) 2N , which solves the SDE j − Y˜ (t)+2N 2/3 N dt dY˜ (t)= dB (t) j dt + , j I . (3.12) j j − 2N 1/3 ˜ ˜ ∈ N k: k=j Yj(t) Yk(t) Xk=16 − u ˜ u ˜ N P˜ξN P˜ξN We denote the distribution of process (Y(t)) with (Y(0)) = ξ by Ai . Process (Ξ(t), Ai ) has N N 1 µ m u− (3.13) Ai ≡ Ai ◦ 14 with 1 1 N mN (dx )= x x 2 exp (x N 1/3)2 dx Ai N Z | i − j| −2 k − N 1 i 1 N 1 N mN (dx )= x x 2 xν+1/2 exp x dx , (3.15) Jν N Z | i − j| j −2 k N 1 i N N 1 µ m u− (3.16) Jν ≡ Jν ◦ N as the reversible measure. Measure µJν is the DPP with correlation kernel N 1 1 − x y K(ν)(x, y)= ϕν ϕν , N 2N k 2N k 2N Xk=0 where ν Γ(k + 1) ν/2 ν x/2 ϕk(x)= x Lk(x)e− , sΓ(ν + k + 1) ν N with the Laguerre polynomials Lk(x),k 0 with parameter ν > 1 [8]. From the construc- P˜ξN ∈ − tion of ⋆ , ⋆ sin, Ai Jν ; ν > 1 the following is readily derived from Proposition 3.3. ∈ { }∪{ − } Proposition 3.4 Let ⋆ sin, Ai J ; ν > 1 . Suppose that ξ X . Then ∈{ }∪{ ν − } ∈ ⋆ l ξN ξ (Ξ(t), P˜⋆ ) (Ξ(t), P ), N , → ⋆ →∞ M l weakly on C([0, ), ), where ξN is the configuration with a finite number of particles given in (2.13). ∞ Remark (i) Let ⋆ sin, Ai J ; ν > 1 . The reversible diffusion process (Ξ(t), P˜ξN ) ∈{ }∪{ ν − } ⋆ is associated with the Dirichlet form ( N , N ) introduced in Section 2. E⋆ D⋆ 15 ν ˜ν (ii) We set Vj (t)= Zj (t). Process V(t)=(Vj(t))j In then solves the SDE ∈ q N ν Vj(t)dt 2ν +1 2Vj(t) dt dVj (t)= dBj(t) + dt + 2 2 , (3.17) − 2N 2Vj(t) Vj(t) Vk(t) k: k=j − Xk=16 where we impose a reflecting wall at the origin for ν ( 1, 0). ξl ξl ∈ − P N P˜ N Pξ (iii) Both processes (Ξ(t), Ai ) and (Ξ(t), Ai ) converge to process (Ξ(t), Ai) as N with the vague topology. However, when considering the limits of the associated SDEs→ (3.5) ∞ t and (3.12), heuristically, they differ by 2 dt. The difference stems from the fact that process PξN (Ξ(t), Ai ) is not reversible and particles far from the origin move in strong drifts. Then ξl process (Ξ(t), P N ) does not converge to (Ξ(t), Pξ ) as N with a stronger topology that Ai Ai →∞ preserves the Markov property. This would be related to the relation between the uniqueness of the strong solution of ISDE and the triviality of the tail σ-field of the path space [27]. Under P , ⋆ = sin, Ai J ,ν > 1 , Ξ C([0, ), M) is a non-explosion and ⋆ { }∪{ ν − } ∈ ∞ non-crossing path and can be naturally lifted to the labeled path X = (Xj)j N such that ∈ R N N Ξ(t)= j N δXj (t) and Xj C([0, ), ), j . We write X Ξ if X Xj j . ∈ ∈ ∞ ∈ ∈ ∈{ } ∈ LemmaP 3.5 (i) Let ⋆ sin, Ai . For each T > 0, there exists a positive constant C⋆ such that for any interval D∈{of R and}ε> 0 ε P⋆ X Ξ s.t. X(0) D, sup X(t) X(0) >ε C⋆(ρ⋆(D) 1) Erf , ∃ ∈ ∈ t [0,T ] | − | ≤ ∨ √T ∈ ! 2 where Erf(a)= ∞ 1 e x /2dx. a √2π − (ii) Let ⋆ J ,ν > 1 . For each T > 0, there exists a positive constant C such that for ν R ⋆ any interval∈{D of [0,− )}with D 1 and ε> 0 ∞ | | ≥ ε P⋆ X Ξ s.t. X(0) D, sup X(t) X(0) >ε C⋆(ρ⋆(D) 1) Erf . ∃ ∈ ∈ t [0,T ] − ! ≤ ∨ √T ∈ p p To prove Lemma 3.5 we prepare a lemma concerning on reversible diffusions in RN . Lemma 3.6 Let Y(t)=(Yj(t))j I be a reversible diffusion process satisfying ∈ N t Y (t)= Y (0) + B (t)+ b (Y(s))ds, j I , t [0, T ], j j j j ∈ N ∈ Z0 with some measurable function b =(b ,...,b ). Here the B (t), 1 j N are independent 1 N j ≤ ≤ BMs, and Y(0) = (Yj(0))j I is a random variable distributed by the reversible probability ∈ N measure, which is independent of the BMs. Putting ρ (A) = N P (Y (0) A) for A Y j=1 j ∈ ∈ (R), and B P ∞ ρ (A ) Erf( 2(ℓ 1) 1 ε/√T ) c = Y ℓ { ∨ − } , (3.18) ρY (D) 1 Erf(ε/√T ) Xℓ=0 ∨ 16 we then obtain ε P Yj(0) D, sup Yj(t) Yj(0) >ε for some j 4(1 + c)(ρY (D) 1) Erf . ∈ t [0,T ] | − | ≤ ∨ √T ∈ ! (3.19) Proof. We use a Lyons-Zheng decomposition for the proof (see for example Section 5.7 in [4]). We deduce from the Lyons-Zheng decomposition that Yˆ (t)= Y(T t) satisfies − t Yˆ (t)= Yˆ (0) + Bˆ (t)+ b (Yˆ (s))ds, j I , t [0, T ], j j j j ∈ N ∈ Z0 with independent BMs Bˆ (t), 1 j N. Then j ≤ ≤ t Y (t) Y (0) = Yˆ (T t) Yˆ (T )= Bˆ (T t) Bˆ (T ) b (Y(s))ds j − j − − j − − j − j Z0 1 = B (t)+ Bˆ (T t) Bˆ (T ) . (3.20) 2 j j − − j Putting Aℓ = Aℓ(ε)= x R : infy D x y [ℓε, (ℓ + 1)ε) , ℓ N 0 , we have { ∈ ∈ | − | ∈ } ∈ ∪{ } P Yj(0) D, sup Bj(t) ε, sup Yj(t) Yj(0) >ε ∈ t [0,T ] | | ≤ t [0,T ] | − | ∈ ∈ ! ∞ P Yj(0) D, sup Bj(t) ε, Yj(T ) Aℓ, sup Yj(t) Yj(0) (ℓ 1)ε ≤ ∈ t [0,T ] | | ≤ ∈ t [0,T ] | − | ≥ ∨ ! Xℓ=0 ∈ ∈ ∞ P Yj(0) D,Yj(T ) Aℓ, sup Bˆj(T t) Bˆj(T ) 2(ℓ 1) 1 ε by (3.20) ≤ ∈ ∈ t [0,T ] − − ≥{ ∨ − } ! ℓ=0 ∈ X ∞ P Yj(T ) Aℓ, sup Bˆj(T t) Bˆj(T ) 2(ℓ 1) 1 ε . ≤ ∈ t [0,T ] − − ≥{ ∨ − } ! ℓ=0 ∈ X Then from this we deduce that P Yj(0) D, sup Yj(t) Yj(0) >ε for some j ∈ t [0,T ] | − | ∈ ! N P Y (0) D, sup Y (t) Y (0) >ε ≤ j ∈ | j − j | j=1 t [0,T ] ! X ∈ N P Y (0) D, sup B (t) >ε ≤ j ∈ | j | j=1 t [0,T ] ! X ∈ N ∞ + P Yj(T ) Aℓ, sup Bˆj(T t) Bˆj(T ) 2(ℓ 1) 1 ε . ∈ t [0,T ] − − ≥{ ∨ − } ! ℓ=0 j=1 ∈ X X 17 Note that Yˆ (0) is independent of Bˆj(t), 1 j N, since Y(0) is independent of B(t), 1 j N. We then have that the right-hand≤ side≤ of the above inequality is bounded by ≤ ≤ N P (Y (0) D) P sup B (t) >ε j ∈ | j | j=1 t [0,T ] ! X ∈ N ∞ + P (Y (T ) A ) P sup Bˆ (T t) Bˆ (T ) 2(ℓ 1) 1 ε j ∈ ℓ j − − j ≥{ ∨ − } j=1 t [0,T ] ! Xℓ=0 X ∈ N N ε ∞ 2(ℓ 1) 1 ε 4 P (Y (0) D) Erf +4 P (Y (0) A )Erf { ∨ − } , ≤ j ∈ √ j ∈ ℓ √ j=1 T j=1 T X Xℓ=0 X ε ∞ 2(ℓ 1) 1 ε =4ρY (D)Erf +4 ρY (Aℓ)Erf { ∨ − } . √T √T Xℓ=0 Here we used the estimate P supt [0,T ] Bj(t) >ε 4Erf(ε/√T ) and the reversibility of ∈ | | ≤ the process. (3.19) follows from this and (3.18) immediately. Proof of Lemma 3.5. We apply Lemma 3.5 to SDEs (3.9), (3.12), and (3.17). By simple calculation, for the solutions of these SDEs, c is a constant independent of D and ε. We then see that there exists a positive constant C′ > 0 such that for ⋆ sin, Ai ⋆ ∈{ } N ε P˜µ⋆ N ⋆ X Ξ⋆ s.t. X(0) D, sup X(t) X(0) >ε C⋆′ (ρ⋆ (D) 1) Erf , ∃ ∈ ∈ t [0,T ] | − | ≤ ∨ √T ∈ ! and for ⋆ J ,ν > 1 ∈{ ν − } N ε P˜µ⋆ N ⋆ X Ξ⋆ s.t. X(0) D, sup X(t) X(0) >ε C⋆′ (ρ⋆ (D) 1) Erf , ∃ ∈ ∈ t [0,T ] − ! ≤ ∨ √T ∈ p p N N N N where ρ⋆ (D) = D ρ⋆ (x)dx with the density (the first correlation function) ρ⋆ (x) of µ⋆ . N P˜µ⋆ P M Since ⋆ convergesR to ⋆ weakly on C([0, ), ), as N (see for instance [8, Section 7]), we obtain the desire result by simple observation.∞ →∞ 3.4 Proof of Theorem 2.1 Let X be a subset of M and (Ξ(t),P ), ξ X be a stochastic process. Suppose that ξ ∈ P ( ) P ( ), if ξ ξ in X. (3.21) ξn · → ξ · n → Further suppose ζ X satisfying P (Ξ(t) is continuous in X)=1. If(Ξ(t),P ) is Markovian, ∈ ζ ζ then it is strong Markovian. Let ⋆ sin, Ai J ; ν > 1 . From Proposition 3.3 (1) we see that (Ξ(t), P ) satisfies ∈{ }∪{ ν − } ⋆ (3.21). Hence from the Markov property of process (Ξ(t), P⋆) given in [12] and Proposition 3.3 (ii), we deduce Theorem 2.1 from the following proposition. 18 Proposition 3.7 Let ⋆ sin, Ai J ; ν > 1 . Then process (Ξ(t), P ) is continuous ∈ { }∪{ ν − } ⋆ in X⋆. Proof. This proposition is derived from the following claims: 1) Ξ(t) has a vaguely continuous path. 2) For each T N, there exist ε (0, 1), κ (1/2, κ ) such that ∈ ∈ ∈ ⋆ P Xε,κ lim lim ⋆ Ξ(t) L,m, t [0, T ] =1. L m →∞ →∞ ∈ ∈ Claim 1) has already shown in Section 3.2. We remark that it is also derived by Kolmogorov’s criterion: for any polynomial function f, which is a smooth function on M givne in (3.27), E [ f(Ξ(t)) f(Ξ(s)) β] C t s α, 0 s < t T < , ⋆ | − | ≤ | − | ≤ ≤ ∞ for some α> 1,β > 0, and C > 0, which is readily proved from Lemma 3.5. Claim 2) is derived from two estimates ε lim P⋆ ( ρ⋆(DL) Ξ(t, DL) L , t [0, T ]) = 1,DL = [0, L], [ L, 0], (3.22) L →∞ | − | ≤ ∈ − and κ κ lim P⋆ (Ξ(t, [g (k),g (k + 1)]) m, t [0, T ] k Z)=1. (3.23) m →∞ ≤ ∈ ∈ From Lemma 3.1, there exist m′ N and p < m′ 1 such that ∈ − m′ p µ⋆(dξ) ρ⋆(DL) ξ(DL) = (L ), L . (3.24) M − O →∞ Z Taking ε ((p + 1)/m′, 1) and using Chebyshev’s inequality with (3.24), we can find a positive constant∈ C such that ε p m′ε µ ( ρ (D ) ξ(D ) L ) CL − . ⋆ | ⋆ L − L | ≥ ≤ Since p m′ε< 1, we have − − ∞ µ ( ρ (D ) ξ(D ) ) Lε) < . | ⋆ L − L | | ≥ ∞ L=1 X By Borel-Cantelli’s lemma and the stationarity of the process, for any k N ∈ ε j lim P⋆ ρ⋆(DL) Ξ(t, DL) L , t = , j =1, 2,...,kT =1. L | − | ≤ k →∞ (3.22) is then derived from Lemma 3.5. Estimate (3.23) is derived from Lemmas 3.2 and 3.5. In fact Lemma 3.5 (i) implies that for ⋆ sin, Ai and k Z, ∈{ } ∈ κ κ κ 1 P⋆ X Ξ s.t. X(0) [g (k),g (k + 1)], sup X(u) X(0) > k − ∃ ∈ ∈ u [0,1/kℓ] | − | | | ∈ ! κ κ κ+ℓ/2 1 C ρ ([g (k),g (k + 1)]) 1 Erf( k − ). ≤ ⋆{ ⋆ ∨ } | | 19 Take ℓ 2 and choose m N such that (3.1) holds. Using Borel-Cantelli’s lemma with ≥ ∈ simple calculations, we see that for P⋆-a.s. Ξ, there exists k0 = k0(Ξ) N such that for any Z j ℓ ∈ k with k k0, if X Ξ and s k ℓ ; j =1, 2,..., k T satisfy ∈ | | ≥ ∈ ∈{ | | | | } X(s) [gκ(k),gκ(k + 1)], ∈ then κ 1 sup X(s + u) X(s) k − . u [0,1/kℓ] | − |≤| | ∈ Combining this fact with (3.2), we see that for P⋆-a.s. Ξ there exists m1 = m1(Ξ) N such that ∈ Ξ(t, [gκ(k),gκ(k + 1)]) m , , t [0, T ], k Z. (3.25) ≤ 1 ∈ ∈ We thus obtain (3.23). For ⋆ jν,ν > 1 we can obtain the desire result by the same argument as above with Lemma∈ 3.5 { (ii). This} completes the proof. 3.5 Proof of Theorem 2.2 We now introduce Dirichlet forms describing k-labeled dynamics. For this we recall the definition of Palm and Campbell measures. Let x =(x ,...,x ) Rk. We set k 1 k ∈ k µx = µ( δ ξ(x ) 1 for i =1,...,k), k · − xi | i ≥ i=1 X which is called the (reduced) Palm measure of µ. The Campbell measure for probability measure µ is then given by k ν (dxkdη)= µxk (dη)ρk(xk)dxk. k Here ρk : R [0, ) is the k-th correlation function of µ and dxk = dx1 dxk is the → ∞Rk k Rk k ··· k Lebesgue measure on as before. Let 0 = C0∞( ) . For f,g 0 , let [f,g] be D ⊗D∞ ∈D ∇ such that 1 k ∂ ∂ k[f,g](x ,ξ)= a(x ) f(x ,ξ) g(x ,ξ). (3.26) ∇ k 2 j ∂x k ∂x k j=1 j j X We set Da,k as Da,k[f,g][x ,ξ]= k[f,g](x ,ξ)+ D[f(x , ),g(x , )](ξ). k ∇ k k · k · a,k νk 2 Rk M k We consider the bilinear form ( νk , ) on L ( , ν ) defined as E D∞ × a,k Da,k k νk (f,g)= (f,g)dν , E M Z νk k a,k 2 = f 0 : νk (f, f)+ f L2(Rk M,νk) < . D∞ { ∈D E k k × ∞} 20 a,k νk a,k νk It was proved that ( νk , ) is closable and its closure ( νk , ) is a quasi-regular Dirichlet E D∞ k E D form in case µ is a quasi Gibbs measure [21]. Let ν⋆ be the Campbell measure associated k k with DPP µ⋆, ⋆ sin, Ai Jν ; ν > 1 . For the case where (ν , a(x)) = (νsin, 1), k k ∈ { }∪{ − } k k (νAi, 1), and (νJ , 4x), we denote the quasi-Dirichlet form by ( ⋆ , ) and the associated ν x E D diffusion process by ((Xk(t),H(t), P k,η), ⋆ sin, Ai J ; ν > 1 . We introduce the ⋆ ∈ { }∪{ ν − } map u : Rk M M defined as × → k u(x ,ξ)= δ + ξ, x Rk, ξ M. k xj k ∈ ∈ j=1 X x u u k k,η It is shown in [21] that if (xk, η)= ξ, process ( (X (t),H(t)), P⋆ ) coincides with process ξ (Ξ(t), P⋆). Let ⋆ sin, Ai J ; ν > 1 . Theorem 2.1 implies that the Dirichlet form ( ˆ , ˆ ) ∈{ }∪{ ν − } E⋆ D⋆ associated with process (Ξ(t), P⋆) is quasi-regular [16, 4]. A function f on the configuration space M is said to be polynomial if it is written in the form f(ξ)= F φ1(x)ξ(dx), φ2(x)ξ(dx),..., φk(x)ξ(dx) (3.27) R R R Z Z Z k with polynomial function F on R ,k N, and smooth functions φj, 1 j k on R with compact supports. Let be the set of∈ all polynomial functions on M.≤ In≤ [8, Proposition P 7.2] we showed that ˆ (f,g)= (f,g), f,g . (3.28) E⋆ E⋆ ∈ P ( ⋆, ⋆) and ( ˆ⋆, ˆ⋆) are then closed extensions of ( ⋆, ), and the former is the smallest one [25].E D These relationsE D are generalized to k-labeled dynamics.E P Lemma 3.8 Let ⋆ sin, Ai Jν ; ν > 1 . For each k N, there exists a diffusion ∈ { x }∪{ − } ∈ process ((Xk(t),H(t)), Pˆ k,η) associated with quasi-regular Dirichlet form ( ˆk, ˆk) such that ⋆ E⋆ D⋆ x u k k,η Pξ u ( (X (t),H(t)), P⋆ ) = (Ξ(t), ⋆), if (xk, η)= ξ, (3.29) k(f,g)= ˆk(f,g) f,g C (Rk) . (3.30) E⋆ E⋆ ∈ 0 ⊗ P Proof. Let ⋆ sin, Ai J ; ν > 1 . We introduce ( ˜k, ˜k ) defined as ∈{ }∪{ ν − } E⋆ D⋆,0 k k ˜ = f(x , η)= g(x , u(x , η)); g C∞(R ) , D⋆,0 { k k k ∈ 0 ⊗D⋆} ˜k k Dk k ˜k ⋆ (f,g)= [f,g](xk, η)+ ⋆[f(xk, ),g(xk, )](η)ν⋆ (dxkdη), f,g ⋆,0, E M ∇ · · ∈ D Z Dk D1,k where the derivatives are taken in the sense of the Schwartz distribution, ⋆ = if Dk D4x,k M N ⋆ sin, Ai , and ⋆ = if ⋆ Jν,ν > 1 . For η and r , we set ∈ { } ℓ ∈ { k − } ∈ ∈ c ˜ ηr = η[ r,r] = δyj and ζ = η[ r,r] . For f ⋆,0 we set − j=1 − ∈ D P fr,ζ (xk, ηr)= f(xk, η)1[ r,r]k (xk) − 21 and Dk [f,g][x , η]= k[f ,g ](x , η )+ Dk[f (x , ),g (x , )](η ) ⋆,r k ∇ r,ζ r,ζ k r ⋆ r,ζ k · r,ζ k · r ˜k Dk k ˜k ⋆,r(f,g)= ⋆,r(f,g)dν , f,g ⋆,0. E M ∈ D Z It is readily seen that the bilinear forms ( ˜k , ˜k ), r N are closable and increasing. From E⋆,r D⋆,0 ∈ [19, Lemma 2.1 (1)] we see that ( ˜k, ˜k ) is closable. The quasi-regularity of the closure E⋆ D⋆,0 ˆk ˆk ˜k ˜k ˆ ˆ ( ⋆ , ⋆ )of( ⋆ , ⋆ ) is derived from that of the Dirichlet form ( ⋆, ⋆). Hence, the associated x E D E D k P k,η E D diffusion process ((X (t),H(t)), ⋆ ) can be constructed. Equation (3.30) is derived from (3.28), while (3.29) is derived from the argument to show [21, Lemma 4.2]. This completes the proof. Proof of Theorem 2.2. We show Theorem 2.2 (i) by applying [22, Theorem 26] to process (Ξ(t), P ) associated with the Dirichlet form ( ˆ , ˆ ), ⋆ sin, Ai J ; ν > 1 . Of the ⋆ E⋆ D⋆ ∈{ }∪{ ν − } assumptions (A.1)–(A.5) of the theorem, (A.1), (A.2), and (A.5) are satisfied, because the related measures µ are the same as those of the Dirichlet form ( , ), which have already ⋆ E⋆ D⋆ been verified in [22, 5, 26]. Assumption (A.4) is derived from the fact that the capacity related to ( ⋆, ⋆) is greater than that of ( ˆ⋆, ˆ⋆), since these Dirichlet forms are closed extensions ofE ( D, ), and the former is the smallestE D one. E⋆ P Condition (A.3) is used in the proof of [22, Theorem 26] to construct a k-labeled process and to check that process Xj(t) Xj(0), j = 1, 2,...,k can be regarded as a Dirichlet − k ˆk process. These claims are derived from Lemma 3.8 together with the fact that ⋆ ⋆ . (i) is thus proved. D ⊂ D Claim (ii) is derived from (i) and the uniqueness of the strong solutions of ISDEs (sin), (Ai), and (Jν), which have been proved in [27]. ξN Proof of Corollary 2.3. 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