Uniqueness of Dirichlet forms related to infinite systems of interacting Brownian motions

The Dirichlet forms related to various infinite systems of interacting Brownian motions are studied. For a given random point field $ \mu $, there exist two natural infinite-volume Dirichlet forms $ (\E^{\Os},\d^{\Os})$ and $(\E^{\La},\d^{\La})$ on $ \Lm $ describing interacting Brownian motions each with unlabeled equilibrium state $ \mu $. The former is a decreasing limit of a scheme of such finite-volume Dirichlet forms, and the latter is an increasing limit of another scheme of such finite-volume Dirichlet forms. Furthermore, the latter is an extension of the former. We present a sufficient condition such that these two Dirichlet forms are the same. In the first main theorem (\tref{l:31}) the Markovian semi-group given by $(\E^{\La},\d^{\La})$ is associated with a natural infinite-dimensional stochastic differential equation (ISDE). In the second main theorem (\tref{l:32}), we prove that these Dirichlet forms coincide with each other by using the uniqueness of solutions of ISDE. We apply the result to stochastic dynamics arising from random matrix theory such as the sine, Bessel, and Ginibre interacting Brownian motions and interacting Brownian motions with Ruelle's class interaction potentials.

1 Introduction An infinite system of interacting Brownian motions in R d can be represented by (R d ) Nvalued stochastic process X = (X i ) i∈N [10,11,14,18]. This process is realized using several probabilistic constructs such as stochastic differential equation, Dirichlet form theory, and martingale problems. Among them, the second author constructed in a general setting processes using the technique of Dirichlet forms [14,18]. Specifically, the Dirichlet form introduced, (E upr , D upr ) is obtained by the smallest extension of the bilinear form (E µ , D µ • ) on L 2 (S, µ) with domain D µ • defined by E µ (f, g) = S D[f, g](s) µ(ds), where D • is the set of all local smooth functions on the (unlabeled) configuration space S introduced in (2.1),f is a symmetric function such thatf (s 1 , s 2 , . . . ) = f (s), · is the inner product in R d , and s = i δ s i denotes a configuration. If we take µ to be the Poisson random point field, the intensity of which is the Lebesgue measure, then the diffusion given by the Dirichlet form (E upr , D upr ) is the unlabeled Brownian motion B such that is a system of independent copies of the standard Brownian motion.
This Dirichlet form is a decreasing limit of Dirichlet forms associated with finite systems of interacting Brownian motions in bounded domains S R = {x ∈ R d ; |x| ≤ R} with a boundary condition. Because of the boundary condition, Brownian particles that touch the boundary disappear. Also, particles enter the domain from the boundary according to the reversible measure µ.
In contrast, Lang constructed the infinite system of Brownian motions as a limit of stochastic dynamics in bounded domains S R by considering finite systems with another boundary condition [10,11]. In his finite systems, a particle hitting the boundary is reflected and hence the number of particles in the domain is invariant. His process is associated with the Dirichlet form (E lwr , D lwr ) that is the increasing limit of the Dirichlet forms associated with finite systems with the reflecting boundary condition.
In this paper, we discuss the relation between these Dirichlet forms, (E upr , D upr ) and (E lwr , D lwr ). The main purpose of this paper is to give a sufficient condition for (E lwr , D lwr ) = (E upr , D upr ). (1.1) By construction the inequality (E lwr , D lwr ) ≤ (E upr , D upr ) (1.2) always holds whereas (1.1) does not necessarily hold in general. Although the problem is quite natural and general, little is known about the equality (1.1). To the best of our knowledge, the unique example for which the equality (1.1) holds is the system of hard-core Brownian balls proved by the third author [26].
The study of infinite systems of interacting Brownian motions was initiated by Lang [10,11] and continued by Fritz [3], the third author [25], and others. In their respective work, the free potential Φ is assumed to be zero and the interaction potentials Ψ are of class C 3 0 (R d ) or exponentially decay at infinity and satisfying the super-stability in the sense of Ruelle. The infinite-dimensional stochastic differential equation (ISDE) is given by Here β > 0 is an inverse temperature. Lang constructed a solution for the µ-a.s. x unlabeled initial point, where µ is a grand canonical Gibbs measure with interaction potential Ψ. Indeed, Lang and others solved the ISDE as a limit of solutions of finite-volume stochastic differential equations (SDE), describing particles in S R with reflecting boundary condition on ∂S R . That is, the SDE is given by with the initial condition X 0 = (x i ) ∞ i=1 such that |x i | < |x i+1 | for all i ∈ N, and x(S R ) coincides with the number of particles in S R . The process L R,i = {L R,i t } denotes the local time-type drift arising from the reflecting boundary condition on ∂S R (see (2.24) for L R,i ) and n R (x) is the inward normal vector at x ∈ ∂S R .
In contrast, the labeled diffusion in S R introduced in [14] is given by the SDE (1.5) with the foregoing boundary condition. These SDEs have thus different boundary conditions. The solutions of (1.4) are non-ergodic, whereas the solutions of (1.5) are ergodic. Indeed, the system in (1.4) keeps the initial number of particles in S R . In the second dynamics, the number of particles in S R varies. The state space of solutions in (1.5) therefore consists of a unique ergodic component (regarded as { ∞ m=0 (S int R ) m }-valued process, where (S int R ) 0 = {∅} and S int R is the interior of S R ). Let (E lwr R , D lwr R ) be the Dirichlet form on L 2 (S, µ) associated with (1.4), that is, the Dirichlet form (E lwr R , D lwr R ) describes the motion of unlabeled dynamics related to (1.4). Let (E upr , D upr R ) be the Dirichlet form on L 2 (S, µ) associated with (1.5). Here we use the notation (E upr , D upr R ) rather than (E upr R , D upr R ) because (E upr , D upr R ) is the closure of (E, D µ • ∩ B R (S)) (see Lemma 2.1 see for notational details), whereas (E lwr R , D lwr R ) is the closure with respect to the energy form E R different from E. As we shall see later, these two Dirichlet forms satisfy the relation } is an increasing scheme of Dirichlet forms, whereas {(E upr , D upr R )} is decreasing. This fact implies the obvious relation (1.2). The difference in these schemes lies in the boundary condition. Therefore, our task is to control the effect of the boundary condition to prove it becomes negligible as R → ∞.
The main examples of our models have a logarithmic interaction potential, which is a very long rang potential that has quite strong long-range effect. We emphasize that the ISDEs arising from random matrix theory usually have logarithmic interaction potentials, and hence this class of interacting Brownian motions is significant.
The typical ISDE for logarithmic potentials is the Ginibre interacting Brownian motion in R 2 with the ISDE and .
Surprisingly, these two ISDEs have the same solution that defines the Ginibre interacting Brownian motion. This is a consequence of the long-rang effect of the logarithmic interaction potential whereby the motion of the particles is suppressed very strongly.
Our result proves nevertheless the uniqueness of Dirichlet forms for which (1.1) holds, a phenomenon similar to short-range interaction potentials.
In the first main theorem (Theorem 3.1), we shall prove that any limit point of the solutions of (1.4) is a solution of the ISDE (1.3) satisfying well-behaved properties (see Theorem 3.1). The limit points of the solutions of (1.5) were proved to satisfy the ISDE (1.3) with the same well-behaved properties [14,17]. Hence, assuming the uniqueness of solutions of (1.3) under the foregoing well-behaved properties, these two limits of the solutions are the same. This establishes the coincidence of the two Dirichlet forms (E lwr , D lwr ) and (E upr , D upr ).
The motivation of our work lies in the recent rapid and outstanding progress of random matrix theory in proving that the random point fields arising from Gaussian random matrices (invariant random matrices) such as sine β , Airy β , and Ginibre random point fields, are universal. Indeed, these random point fields are obtained as scaling limits of eigenvalue distributions of a quite general class of random matrices and also log gases with general free potentials. Once this static universality is established, it is natural to pursue its dynamical counter part. In a forthcoming paper, the first and second authors will prove that the natural reversible stochastic dynamics associated with these random point fields are also universal objects. Examples of universal stochastic dynamics are the sine, Airy, and Ginibre interacting Brownian motions (see Section 8). They are limits of the stochastic dynamics related to N-particle systems with reversible random point fields that converge to those universal random point fields mentioned above. This result is established by assuming the limits of the lower and upper Dirichlet forms in (1.1) are equal in addition to a certain strong convergence of the random point fields. Hence our main theorem (Theorem 3.2) plays a crucial role in the dynamical universality of random matrices in the sense given above.
The organization of the paper is as follows: In Section 2, we prepare the two schemes of the Dirichlet forms describing interacting Brownian motions, and quote related results. In Section 3, we state the main theorems (Theorem 3.1 and Theorem 3.2). In Section 4, we prove Theorem 3.1. In Section 5, we prove Theorem 3.2. In Section 6, we comment on a generalization to the uniformly elliptic case. In Section 7, we construct cut-off coefficients b r,s,p appearing in (A6). In Section 8, we present examples. In Appendix (Section 9) we present a set D • used in Section 2.4 and prove Lemma 2.5. In Section 10, we give concluding remarks with some open questions.

Two schemes of Dirichlet forms
Let S be a closed set in R d with interior S int which is a connected open set satisfying S int = S and the boundary ∂S having Lebesgue measure zero.
A configuration s = i δ s i on S is a Radon measure on S consisting of delta masses δ s i . Let S be the configuration space over S. Then, by definition, S is the set given by (2.1) By convention, we regard the zero measure as an element of S. We endow S with the vague topology, which makes S to be a Polish space. A probability measure µ on (S, B(S)) is called a random point field on S. We assume µ is supported on the set consisting of infinitely-many particles: Let S r = {s ∈ S ; |s| ≤ r} and S m r = S r × · · · × S r be the m-product of S r . Let S m r = {s ∈ S ; s(S r ) = m} for r, m ∈ N. We set maps π r , π c r : S → S such that π r (s) = s(· ∩ S r ) and π c r (s) = s(· ∩ S c r ).
. For a function f : S → R we set f m r (s, x) = f m r,s (x) such that f m r : S × S m r → R and that f m r,s , with r, m ∈ N, satisfies (1) f m r,s is a permutation invariant function on S m r for each s ∈ S.
When f is σ[π r ]-measurable, the S m r -representations are independent of s. In this case we often write f m r instead of f m r,s . We set Note that D • ∩ L 2 (S, µ) is dense in L 2 (S, µ) and Moreover, we set Note that D m r [f, g] is independent of the choice of the S m r -coordinate x m r (s) and is welldefined. We now define bilinear forms on D • : We note that E µ r (f, f ) is nondecreasing in r, and hence the limit in (2.5) exists. We assume We present later a sufficient condition regarding (2.6); see (A1) in Section 2.2. (1) (E µ , D µ • ∩ B r (S)) and (E µ r , D µ • ) are closable on L 2 (S, µ) for each r.

Quasi-Gibbs measures, unlabeled diffusions, and labeled dynamics
Let Λ r be the Poisson random point field whose intensity is the Lebesgue measure on S r and set Λ m r = Λ r (· ∩ S m r ). Let Φ : S → R ∪ {∞} and Ψ : S 2 → R ∪ {∞} be measurable functions such that Ψ(x, y) = Ψ(y, x). Following [18,19] we quote: Definition 2.1. A random point field µ is called a (Φ, Ψ)-quasi Gibbs measure if its regular conditional probabilities µ m r,s = µ( π r (x) ∈ · | π c r (x) = π c r (s), x(S r ) = m) satisfy, for all r, m ∈ N and µ-a.s. s, Here c 1 = c 1 (r, m, s) is a positive constant depending on r, m, s. For two measures µ, ν on a σ-field F , we write µ ≤ ν if µ(A) ≤ ν(A) for all A ∈ F . Moreover, H m r is the Hamiltonian on S r defined by Here Z m r,s is the normalizing constant. For random point fields appearing in random matrix theory, interaction potentials are logarithmic functions, where the DLR equations do not make sense as stated because the term x i ∈Sr, s j ∈S c r Ψ(x i , s j ) diverges. The notion of a quasi-Gibbs measure still makes sense for logarithmic potentials.
We make the following assumption.
If these interaction potentials are translation invariant, we often write Ψ(x, y) = Ψ(x− y) andΨ(x, y) =Ψ(x−y). The importance of (A1) lies in the fact that it gives a sufficient condition of the basic assumption (2.6). We quote: We now recall two basic notions on random point fields: correlation functions and density functions.
A symmetric and locally integrable function ρ n : S n → [0, ∞) is called the n-point correlation function of a random point field µ on S with respect to the Lebesgue measure if ρ n satisfies for any sequence of disjoint bounded measurable sets A 1 , . . . , A m ∈ B(S) and a sequence of positive integers k 1 , . . . , k m satisfying k 1 +· · ·+k m = n. When s(A i )−k i < 0, according to our interpretation, s(A i )!/(s(A i ) − k i )! = 0 by convention. We assume that µ has n-point correlation function ρ n for each n ∈ N.
A symmetric function σ k r : S k r → [0, ∞) is called the k-point density function of a random point field µ on S r with respect to the Lebesgue measure if for all non-negative, bounded Let S m r = {s ∈ S ; s(S r ) = m} as before. We make the following assumption.
Assume (A1). Then we deduce from Lemma 2.1 and Lemma 2.4 that the non-negative . Therefore, let (E upr , D upr ) be its closure on L 2 (S, µ). The next result is a refinement of [14, 119p. Corollary 1] and can be proved in a similar fashion. We postpone the proof to Appendix (see Section 9.2). We note that (A2) is used to guarantee the existence of the diffusion. The µreversibility of the diffusion follows from 1 ∈ D upr and symmetry of (E upr , D upr ).
By construction, such a family of diffusion measures P upr = {P upr x } with quasi-continuity in x is unique for quasi-everywhere starting point x. Equivalently, there exists a set S 0 such that the complement of S 0 has capacity zero and the family of diffusion measures P upr = {P upr x } associated with the Dirichlet space above with quasi-continuity in x is unique for all x ∈ S 0 and P upr x (X t ∈ S 0 for all t) = 1 for all x ∈ S 0 . We next lift the unlabeled dynamics X to a labeled dynamics We call X i tagged particles and X = (X i ) i∈N labeled dynamics. Note that for a given unlabeled process X, there exist plural labeled dynamics in general. We next give a condition such that X = (X i ) i∈N is determined uniquely. For this purpose, we impose the following condition: x }, each tagged particle {X i } i∈N does not collide with another. Furthermore, {X i } i∈N does not hit the boundary ∂S of S. This condition is equivalent to both the capacity of multiple points and that of configurations with particles on the boundary ∂S being zero: Here Cap µ denotes the one-capacity with respect to the Dirichlet space (E µ , D µ ) on L 2 (S, µ) (see [4] for the definition of capacity). Let 2π)e −|x| 2 /2 dx be the error function. We further assume: From (A4), we deduce the non-explosion of each tagged particle [16, Theorem 2.5]. We hence see from (A3) and (A4) that under P upr = {P upr x } each tagged particle of {X i } i∈N neither collide each other nor hit the boundary ∂S nor explode.
We call u the unlabeling map if u((x i )) = i δ x i . We call l a label if l : S → S N is a measurable map defined for µ-a.s. x, and u • l(x) = x. For simplicity, we take l as throughout the paper. Because µ has an m-point correlation function for each m, l(x) is well defined for µ-a.s. x.
Lemma 2.6 ( [16,18]). Assume that (A1)-(A4). Let l be a label. Then under P upr = {P upr x } there exists a unique, labeled dynamics X = (X i ) i∈N ∈ C([0, ∞); S N ) such that X 0 = l(X 0 ) and that X t = i∈N δ X i t for all t. Once the initial label l is assigned, the particles are marked forever because they neither collide nor explode. We hence determine the labeled dynamics X from the unlabeled dynamics X and the label l uniquely. We have thus had a natural correspondence between X and (X, l) under the conditions (A3) and (A4). We remark here that X t = l(X t ) for t > 0 in general.
The next lemma will be used in the proof of Lemma 2.9 and Lemma 4.2.

ISDE-representation: Logarithmic derivative
We next present the ISDE describing the labeled dynamics given by Lemma 2.6. The key notion for this is the logarithmic derivative of µ to be introduced below. We first recall two new measures arising from random point field µ. The first concerns the conditioning of µ, the second its disintegration. For We now recall the notion of the logarithmic derivative of µ [17].
We make the following assumption: (A5) µ has a logarithmic derivative d µ . The next lemma reveals the importance of logarithmic derivative.
Then there exists S 0 ⊂ S such that µ(S 0 ) = 1 and that the labeled dynamics X = (X i ) i∈N under P upr x solves the ISDE for each x ∈ S 0 Let X = (X i ) i∈N be a solution of ISDE (2.13) and denote by X t = ∞ i=1 δ X i t the associated unlabeled process. Let µ t be the distribution of X t . We make the following assumptions on X.
(µ-AC) The µ-absolutely continuous condition is satisfied. That is, if X 0 = µ in law, then where µ t ≺ µ means µ t is absolutely continuous with respect to µ.
(NBJ) The no-big-jump condition is satisfied. That is, if the distribution of X 0 equals to µ, then for each r, T ∈ N P (I r,T (X) < ∞) = 1, (2.16) where I r,T is the maximal label with which the particle intersects S r defined by . Then under P upr = {P upr x } x∈S the labeled dynamics X satisfies the conditions (µ-AC), and (NBJ).
Proof. Because the unlabeled dynamics X is µ-reversible, µ t = µ for all t. Hence (µ-AC) is obvious. The second claim follows from the Lyons-Zheng decomposition and Lemma 2.7 (see [21,Lemma 9.4] for detail).

Finite systems in S R of interacting Brownian motions with reflecting boundary condition
We give the SDE representation of the unlabeled process X associated with the Dirichlet form (E lwr R , D lwr R ) on L 2 (S, µ). We denote by P lwr Then from (2.18) we see that the capacity of (E lwr R , D lwr R ) is dominated by that of (E upr , D upr ). Hence non-collision of tagged particles under P lwr R follows from that of the limit diffusion X given by (E upr , D upr ), which is assumed by (A3). With the same reason, tagged particles under P lwr R do not hit the set (∂S) ∩ S R . Non-explosion of tagged particles under P lwr R is obvious because they are reflecting diffusion on S R and frozen outside S R .
We now denote by X = (X i ) ∞ i=1 the labeled process associated with X and the label l.
By definition X 0 = l(X 0 ). The process X under P lwr R describes the system of interacting Brownian motions in which 1. each particle in S R moves in S R and when it hits the boundary ∂S R , it reflects and enter the domain S R immediately, 2. the particles out of S R stay the initial positions forever.
We denote by µ Rs the regular conditional distribution defined by Then µ Rs is a probability measure supported on S Rs . Let µ Rs, [1] be the 1-Campbell measure of µ Rs . Then we have µ Rs, [1] where ρ Rs,1 is the one-point correlation function of µ Rs and µ Rs x is the reduced Palm measure of µ Rs conditioned at x. By the Green formula, we see for all where S R is the Lebesgue surface measure on the boundary ∂S R and n R (x) is the inward normal unit vectors at x ∈ ∂S R . Hence for µ-a.s. s and for µ Rs x -a.s. y, the logarithmic derivative d Rs of µ Rs coincides with the sum of d µ (x, π R (y) + π c R (s)) for x ∈ S R and a singular part associated with the the boundary ∂S R . We then obtain informally Here we naturally extend the domain of d Rs (x, y) to S × S by taking d Rs (x, y) = 0 for x ∈ S R . This is reasonable because particles outside S R are fixed.
By definition x(S R ) coincides with the number of particles in S R for a given configuration x. From the Green formula (2.21) we see that X = (X i ) ∞ i=1 is the system of infinite number of particles such that only particles in S R move and satisfies the following SDE: For µ-a.s. s = i δ s i and for µ Rs -a.s.
and L R,i = {L R,i t } are non-negative increasing processes; see for instance [1]. The particles outside S R are frozen by (2.25).
Then we can easily deduce from (A1) and (A2) that (E lwr R , D lwr R ) is a quasi-regular Dirichlet form on L 2 (S, µ) and that there exists the associated diffusion X. The capacity for (E lwr R , D lwr R ) is dominated by that for (E upr , D upr ). Hence from (A3) we deduce that X has also non-collision property. Clearly, each tagged particle of X does not explode because of the definition of E lwr R . We have thus obtained the labeled process X from the unlabeled process X and the label l. Moreover, using the Fukushima decomposition and taking (2.22) into account, we see that X is a solution of SDE (2.23)-(2.26).
There exists a countable subset D • of L 2 (S, µ), D lwr R , and D Rs,lwr R for µ-a.s. s such that D • is dense in L 2 (S, µ), D lwr R , and D Rs,lwr R for µ-a.s. s with respect to L 2 (S, µ)-norm, E lwr R,1norm, and E Rs,lwr R,1 -norm for µ-a.s. s, respectively. (see Section 9). Here E lwr R,1 -norm of f is given by E lwr Then we see for Replacing f with T lwr R,t f in (2.34) and using (2.32) we have ) on L 2 (S, µ Rs ). We note here f = f Rs for µ Rs -a.s. Therefore, for µ-a.s. s, Here we used µ Rs (S Rs ) = 1, where S Rs = {y ∈ S ; π c R (y) = π c R (s)} as before. The solutions X lwr and X Rs,lwr are associated with the semi-groups T lwr R,t and T Rs,lwr R,t , respectively. Hence from (2.39) we deduce that these are equivalent in distribution. We thus see that the solutions of SDE (2.23)-(2.26) given by these Dirichlet forms are the same.

Statements of the main results
We present our main results. Let T (S) be the tail σ-field of the configuration space S: Let µ s be a regular conditional probability conditioned by the tail σ-field defined as µ s (·) = µ(·|T (S))(s). where µ Rs, [1] is the one-Campbell measure of µ Rs . In Section 7, we shall present a sufficient condition of (A6) for coefficients b = 1 2 d µ given by a logarithmic derivative d µ with pair interaction Ψ such that Ψ(x, y) = Ψ(y, x) = Ψ(x − y) and inverse temperature β > 0. We assume b ∈ L p loc (S × S, µ [1] ) with p > 1 and Here y = i δ y i , Ψ ∈ C ∞ (R d \{0}), and ̺ s are constants. We then take b r,s,p as follows: where χ r , υ p , and ̟ a[r] are functions defined by (7.1), (7.2), and (7.5), respectively. Let P R l(x) be the distribution of the solution of SDE (2.23)-(2.26) given by the Dirichlet form (E lwr R , D lwr R ) on L 2 (S, µ). The first main theorem of this paper is the following.
For µ-a.s. x, the process X = (X i ) i∈N under P ∞ l(x) is a solution of the ISDE satisfying conditions (µ-AC) and (NBJ). Furthermore, X = (X i ) i∈N under P ∞ l(x) is associated with the resolvent {G lwr α } of the Dirichlet form (E lwr , D lwr ) on L 2 (S, µ) in the following sense: Because (E lwr , D lwr ) on L 2 (S, µ) is a Dirichlet form, there exists the associated Markovian semi-group on L 2 (S, µ) whose resolvent is G lwr α in Theorem 3.1. We have however not yet constructed the associated diffusion. Only a stationary Markov process is thus constructed at this stage. In general, we have to prove quasi-regularity of the Dirichlet form (E lwr , D lwr ) on L 2 (S, µ) for the existence of the associated diffusion (see [12] for quasi-regularity).
The next theorem establishes the existence of the associated diffusion by proving the identity between the Dirichlet forms (E lwr , D lwr ) and (E upr , D upr ).
We introduce another Dirichlet form (E + , D + ). Recall that b = 1 2 d µ by (2.12), where d µ is the logarithmic derivative of µ defined by (2.11). Put Let us denote the distributional derivative f ′ by D x f and set [1] (dxdy), f, g ∈ D + .
• In [21, Theorem 5.3 (2)], it was proved that the following uniqueness of solutions holds, which is enough for Theorem 3.2. We shall use this in Section 8. For P upr µ -a.s. X with label l, we set X m * = {X m * t } is such that X m * t = j>m δ X j t . Note that X m * is determined by m ∈ N, X, and l. Let S sde be a subset of S such that b is well defined on {(s, y); δ s + y ∈ S sde }. We set . By definition the drift coefficient b m,i in (3.13) is timeinhomogeneous and is given by where y = (y 1 , . . . , y m ) ∈ S m . We emphasize the importance of (3.14). The function b is not defined on the whole space S × S. Hence we have to restrict the state space of the associated unlabeled dynamics as S sde . The set H is the totality of the initial starting points, which are not necessarily equal to S sde . Following [21], we introduce: (IFC) There exists (H, S sde ) such that SDE (3.13) and (3.14) has a pathwise unique strong solution for each y m ∈ H m for each m ∈ N and for P upr µ -a.s. X.
Feasible sufficient conditions for (IFC) were given in [21,Section 9.3]. The importance of (IFC) is that it yields the pathwise uniqueness of solutions of ISDE (2.13)-(2.14) together with (µ-AC), (NBJ), and (TT) [21,Theorem 5.3 (2)]. All determinantal random point fields are tail trivial [20]. Hence (TT) is satisfied for sine 2 , Airy 2 , Bessel 2 , and Ginibre random point fields. We now quote a result from [21]. Then these solutions are pathwise unique for µ-a.s. x. That is, for µ-a.s. x, if there exist two such solutions X and X ′ defined on the same probability space with X 0 = X ′ 0 = l(x), then P (X t = X ′ t for all t) = 1. We next consider the case that µ is not tail trivial. We recall the decomposition of µ into µ s given by (3.1)-(3.2). Then it is known that, if (A1) is satisfied, then µ s is tail trivial for µ-a.s. s [21, Lemma 13.2]. (2) Suppose that for µ-a.s. s, ISDE (2.13)-(2.14) has solutions for µ s -a.s. x satisfying (µ s -AC), (NBJ), and (IFC). Then these solutions are pathwise unique. That is, for µ-a.s. s, if there exist two such solutions X and X ′ defined on the same probability space with X 0 = X ′ 0 = l(x) for µ s -a.s. initial starting points x, then P (X t = X ′ t for all t) = 1 for µ s -a.s. x.
In [21], it was proved that solutions of ISDE in Lemma 3.5 (2) satisfy (IFC) if coefficients of ISDE comes from interaction potentials which are smooth outside the origin. See Lemma 9.7 and Section 10 in [21] for details. We then deduce that, even if µ is not tail trivial, (3.11) holds for µ s such that µ s satisfies the conditions mentioned in Lemma 3.5.

Proof of Theorem 3.1
In this section we prove Theorem 3.1. Let µ s be as in (3.1).
Proof. This lemma follows from disintegration of µ on µ s , and also the disintegration of their correlation functions and density functions, and Fubini's theorem.
be the labeled diffusion process starting at l(x) whose unlabeled process is associated with the Dirichlet form (E Rs,lwr R , D Rs,lwr R ) introduced in Section 2.4.
Recall that x(S R ) equals the number of particles in S R . To clarify the dependence on R and s, we write We set the m-labeled process X Rs, [m] such that It is known [16] that X Rs,[m] is the diffusion process associated with the Dirichlet form .
Because the coordinate function [m] is an additive functional of the m-labeled diffusion X Rs,[m] (see [4] for additive functional). We remark here that the m-point correlation function of µ Rs vanishes outside S R .
Applying the Fukushima decomposition to f i , the additive functional A can be decomposed as a sum of a unique continuous local martingale additive functional M Rs,i and an additive functional of zero energy N Rs,i : We refer to [4,Theorem 5.2.2] for the Fukushima decomposition.
We recall another decomposition of A  where I r,T is defined by (2.17). July 26, 2018 Proof. From (4.3), we obtain From (4.8) we easily obtain (4.5).
From the conditions above we have the following lemma.
We say a sequence of random variables is tight if for any subsequence we can choose a subsequence that is convergent in law. We also remark that tightness in C([0, T ]; S N ) for all T ∈ N is equivalent to tightness in C([0, ∞); S N ) because we equip C([0, ∞); S N ) with a compact uniform norm. Recall that b r,s,p ∈ C b (S × S) by (A6). Then tightness of {B Rs,m r,s,p (· ∧ σ Rs,m a )} r,s,p,R∈N follows from (4.14) with a straightforward calculation. We thus obtain (1).
In general, a family of probability measures m a in a Polish space is compact under the topology of weak convergence if and only if for any ǫ > 0 there exists a compact set K such that inf a m a (K) ≥ 1 − ǫ. Using this we conclude (2) from (1) combined with (4.6).
With the same reason as the proof of (1), we obtain (3) from (1) and (2). Here the subscript a in the right hand side of (4.17) denotes the dependence on a. We note that the convergence lim R→∞ L Rs,m (· ∧ σ Rs,m a ) = 0 follows from Lemma 4.3. From Here X s,i in the right hand side is the i-th component of X s = (X s,i ) ∞ i=1 . The same holds for B s,n and we write B s,n = (B s,i ) n i=1 This is the reason why we extend the state space in (3) of Lemma 4.4 from that in (1) and (2). We next check consistency in a in the limits in (4.17) and (4.18). Without loss of generality, we can assume We set X Rs,⋄i Using reversibility of diffusions, we obtain the following dynamic estimates from the static condition (A6). Proof. Let X Rs be the unlabeled diffusion such that X Rs where we set x = i δ x i ∈ S. Then we obtain (4.22) from (3.4) and (4.24).
Let a, Q, R ∈ N be such that a < Q, R. Recall that L Rs,m t = 0 by Lemma 4.3. Then we deduce from (4.25) and Itô-Tanaka formula that  Putting (4.32)-(4.33) together yields We then complete the proof of Theorem 3.1.
For µ-a.s. x, the solutions of ISDE (2.13)-(2.14) satisfying (µ-AC) and (NBJ) are unique in law by assumption. Hence for µ-a.s.x, X upr and X starting at l(x) have the same distribution. Hence the associated semi-group coincides with each other. This together with (3.10) implies (E upr , D upr ) = (E lwr , D lwr ). From Theorem 5.3 we have already obtained (E lwr , D lwr ) = (E + , D + ). Combining these we complete the proof of Theorem 3.2.

Symmetric diffusions for uniformly elliptic differential operators
In this section, we give a remark on a generalization of Theorem 3.1 and Theorem 3.2. For this purpose we introduce a function a : S × S → R d 2 and assume: (B1) a, ∇ x a ∈ C b (S × S), a = t a, and a is uniformly elliptic on S × S. where we set s ⋄i = j =i δ s j for s = i δ s i . Moreover, we set If we replace the square fields D m r and D r in (2.2) and (2.3) with D a,m r and D a r in (6.1) and (6.2) and add assumption (B1), then all results in Section 3 still hold.
7 Construction of cut-off coefficients b r,s,p In this section we construct b r,s,p . For this purpose we prepare cut off functions.
We remark that the construction of b r,s,p as above is robust and can be applied to all examples in this paper.

Examples
We now present examples of random point fields for which the following holds: (E upr , D upr ) = (E lwr , D lwr ) = (E + , D + ), (8.1) which is the claim in Theorem 3.2. In this section we write L p loc (µ [1] ) = L p loc (S × S, µ [1] ), where µ [1] is the one-Campbell measure of µ as before.
Here y = i δ y i and "in L p loc (µ [1] sin,β )" means convergence in L p (S r × S, µ [1] sin,β ) for all r ∈ N. The labeled process X = (X i ) i∈Z solves ISDE: with conditions (µ-AC) and (NBJ) [17,27]. We readily verify conditions (A6) from Section 7. We have checked (A1)-(A6). Moreover, µ sin,2 satisfies (TT) because µ sin,2 is a determinantal random point field [20]. Hence we apply Theorem 3.2 and obtain (8.1) for µ sin,2 and, if β = 1, 4, then for µ s sin,β for µ sin,β -a.s. s. If β = 2, then an algebraic construction of the stochastic dynamics associated with the upper Dirichlet form (E upr , D upr ) was known [7]. The distribution of the dynamics are determined by the space-time correlation functions, which is explicitly given by the concrete determinantal kernel. Because of the identity (E upr , D upr ) = (E lwr , D lwr ) in Theorem 3.2, the same holds for the stochastic dynamics associated with the lower Dirichlet form (E lwr , D lwr ).
The labeled process X = (X i ) i∈N satisfies (A3) solves the ISDE:

Gibbs measures with Ruelle-class potential
Let S = R d with d ∈ N. Let Φ = 0 and we consider ISDE (1.3). Assume that Ψ is smooth outside the origin and is a Ruelle-class potential. That is, Ψ is super-stable and regular in the sense of Ruelle [23]. Here we say Ψ is regular if there exists a positive decreasing function ψ : R + → R and R 0 such that Ψ(x) ≥ −ψ(|x|) for all x, Ψ(x) ≤ ψ(|x|) for all |x| ≥ R 0 , ∞ 0 ψ(t) t d−1 dt < ∞.
Let µ Ψ be canonical Gibbs measures with interaction Ψ satisfying (A2). We do not a priori assume the translation invariance of µ Ψ . Instead, we assume a quantitative condition in (8.4) below, which is obviously satisfied by the translation invariant canonical Gibbs measures. Suppose that, for each p ∈ N, there exist positive constants c 7 and c 8 satisfying for all x such that |x| ≥ 1/p. Here ρ m is the m-point correlation function of µ Ψ . For the non-collision property of tagged particles we assume the following. Suppose that d ≥ 2 or that d = 1 with Ψ is sufficiently repulsive at the origin in the following sense [6]. There exist a positive constant c 9 and a positive function h : (0, ∞) → [0, ∞] satisfying that The labeled process X = (X i ) i∈N solves ISDE: