Infinite-dimensional stochastic differential equations and tail σ -fields II: the IFC condition
aa r X i v : . [ m a t h . P R ] J u l Yosuke Kawamoto, Hirofumi Osada, and Hideki Tanemura Infinite-dimensional stochastic differential equationsand tail σ -fields II: the IFC condition Yosuke Kawamoto , Hirofumi Osada , Hideki Tanemura [MSC 2020] Primary 60K35; Secondary 60H10, 82C22, 60B20
Keywords:
Interacting Brownian motions, Infinite-dimensional stochastic dif-ferential equations, Ransom matrices
Abstract
In a previous report, the second and third authors gave general theoremsfor unique strong solutions of infinite-dimensional stochastic differential equa-tions (ISDEs) describing the dynamics of infinitely many interacting Brownianparticles. One of the critical assumptions is the “IFC” condition. The IFCcondition requires that, for a given weak solution, the scheme consisting of thefinite-dimensional stochastic differential equations (SDEs) related to the ISDEsexists. Furthermore, the IFC condition implies that each finite-dimensionalSDE has unique strong solutions. Unlike other assumptions, the IFC conditionis challenging to verify, and so the previous report only verified solution for so-lutions given by quasi-regular Dirichlet forms. In the present paper, we providea sufficient condition for the IFC requirement in more general situations. Inparticular, we prove the IFC condition without assuming the quasi-regularity orsymmetry of the associated Dirichlet forms. As an application of the theoreticalformulation, the results derived in this paper are used to prove the uniquenessof Dirichlet forms and the dynamical universality of random matrices.
We consider the dynamics of infinitely many interacting Brownian particles in theEuclidean space R d . We assume that each particle X i moves under the effect of itselfand the other infinitely many particles. The dynamics X = ( X it ) i ∈ N can be describedby the following infinite-dimensional stochastic differential equation (ISDE): X it − X i = Z t σ ( X iu , X i ♦ u ) dB iu + Z t b ( X iu , X i ♦ u ) du ( i ∈ N ) . (1.1)Here, B = ( B i ) ∞ i =1 , where { B i } i ∈ N denotes independent copies of d -dimensional Brow-nian motion, and X i ♦ = { X i ♦ t } represents the unlabeled dynamics given by X i ♦ t = ∞ X j = i δ X jt . (1.2) Infinite-dimensional stochastic differential equations and tail σ -fields II The coefficients σ and b are defined on R d × S , where S denotes the configurationspace over R d (see (2.1)). Note that the functions σ and b are independent of i ∈ N and all particles { X jt } j ∈ N ,j = i are indistinguishable in (1.2). These conditions enable(1.1) to describe the motion of identical interacting particles.A pair of ( R d ) N -valued, continuous processes ( X , B ) defined on a filtered proba-bility space (Ω , F , P, {F t } ) satisfying (1.1) is called a weak solution, where B is an {F t } -Brownian motion. Loosely speaking, if X is a functional of B and an initialstarting point s , then the weak solution ( X , B ) is called a strong solution. We saythe pathwise uniqueness of solutions holds if any pair of weak solutions ( X , B ) and( X ′ , B ) with the same Brownian motion B defined on the common filtered probabilityspace (Ω , F , P, {F t } ) with X = X ′ almost surely (a.s.) satisfies P ( X = X ′ ) = 1.We say that uniqueness in law holds if the distributions of X and X ′ coincide for anypair of weak solutions ( X , B ) and ( X ′ , B ′ ) with the same initial distribution. Thepathwise uniqueness of weak solutions implies the uniqueness in law because of theYamada-Watanabe theory [6, 22].Typical examples of ISDEs are interacting Brownian motions. Each particle movesunder the force of its self-potential Φ( x ) and the interaction potential Ψ( x, y ). Then X it − X i = B it − β Z t ∇ x Φ( X iu ) du − β Z t ∞ X j = i ∇ x Ψ( X iu , X ju ) du ( i ∈ N ) . (1.3)Here, ∇ x = ( ∂∂x i ) di =1 and β is a positive constant called the inverse temperature.Lang derived general solutions to ISDE (1.3) by constructing a reversible solutionstarting from almost all points under the condition Φ = 0 and Ψ ∈ C ( R d ) [11, 12].Here, we say that a solution X = ( X i ) i ∈ N is reversible with respect to a randompoint field µ ( µ -reversible) if the associated unlabeled process X = P i ∈ N δ X i is µ -symmetric and µ is an invariant probability of X . Recall that a random point field µ is a probability measure on the configuration space S by definition.Fritz explicitly described the set of starting points for up to four dimensions [3],and the third author of the present paper solved the equation for hardcore Brownianballs [27]. These results used Itˆo’s method, and required the coefficients to be smoothand have compact support. These conditions exclude physically interesting examplesof long-range interaction potentials, such as the Lennard-Jones 6-12 potential andRiesz potentials. In particular, the logarithmic potential that appears in randommatrix theory is also excluded.Typical examples of ISDEs with logarithmic interaction potentials are the Dysonmodel in R and the Ginibre interacting Brownian motion in R : X it − X i = B it + β Z t lim r →∞ ∞ X | X iu − X ju | 0, according to our interpretation, s ( A i )! / ( s ( A i ) − k i )! = 0 by convention.Let ˜ µ [1] be the measure on ( S × S , B ( S ) × B ( S )) determined by˜ µ [1] ( A × B ) = Z B s ( A ) µ ( d s ) , A ∈ B ( S ) , B ∈ B ( S ) . The measure ˜ µ [1] is called the one-Campbell measure of µ . If µ has a one-pointcorrelation function ρ , there exists a regular conditional probability ˜ µ x of µ satisfying Z A ˜ µ x ( B ) ρ ( x ) dx = ˜ µ [1] ( A × B ) , A ∈ B ( S ) , B ∈ B ( S ) . The measure ˜ µ x is called the Palm measure of µ [7].In this paper, we use the probability measure µ x ( · ) ≡ ˜ µ x ( · − δ x ), which is calledthe reduced Palm measure of µ . Informally, µ x is given by µ x = µ ( · − δ x | s ( { x } ) ≥ . We consider the Radon measure µ [1] on S × S such that µ [1] ( dxd s ) = ρ ( x ) µ x ( d s ) dx. We always use µ [1] instead of ˜ µ [1] . Hence, we call µ [1] the one-Campbell measure of µ . Similarly, we define µ [ m ] by µ [ m ] ( A × B ) = Z A × B ρ m ( x ) µ x ( d s ) d x . Here, µ x is the reduced Palm measure of µ conditioned at x ∈ S m . We call µ [ m ] the m -Campbell measure of µ . We set µ [0] = µ and call it the zero-Campbell measure.Note that µ [ m ] is not necessarily a probability measure for m ≥ µ is translation invariant and does not concentrate at theempty configuration, whereas µ [0] = µ is always a probability measure by definition.For a subset A , we set π A : S → S by π A ( s ) = s ( · ∩ A ). A function f on S is saidto be local if f is σ [ π K ]-measurable for some compact set K in S . For such a localfunction f on S and a relatively compact open set O in S such that K ⊂ O , we set osuke Kawamoto, Hirofumi Osada, and Hideki Tanemura 7a function ˇ f = ˇ f O defined on P ∞ k =0 O k such that ˇ f O ( x , . . . x k ) restricted to O k issymmetric in x j ( j = 1 , . . . , k ) for each k and that for x = P i δ x i f ( x ) = ˇ f O ( x , . . . , x k ) . Here, the case k = 0, that is, S corresponds to a constant function. Note that forany relative compact open sets O and O ′ including K ˇ f O ( x , . . . , x k ) = ˇ f O ′ ( x , . . . , x k ) for all ( x , . . . , x k ) ∈ ( O ∩ O ′ ) k . (2.2)Hence, ˇ f O ( x , . . . , x k ) is well defined. We say a local function f is smooth if ˇ f = ˇ f O is smooth in ( x , . . . , x k ) for each k . For a subset A of a topological space, the set consisting of the A -valued continuouspaths on [0 , ∞ ) is denoted by W ( A ) = C ([0 , ∞ ); A ). We equip W ( S N ) with theFr´echet metric dist( · , ∗ ) given bydist( w , w ′ ) = ∞ X T =1 T n ∞ X n =1 n min { , k w n − w ′ n k C ([0 ,T ]; S ) } o for w = ( w n ) n ∈ N and w ′ = ( w ′ n ) n ∈ N , where we set k w k C ([0 ,T ]; S ) = sup t ∈ [0 ,T ] | w ( t ) | .Let e S = { s = P i δ s i } be the set of all measures on S consisting of countable pointmeasures. By definition, S ⊂ e S . Let S = { S ∞ q =0 S q } S S N . Let u : S → e S be such that u (( s i ) i ) = X i δ s i . Then, u ( s ) = s for s = ( s i ) i and s = P i δ s i . Here S is regarded as S = {∅} and u ( ∅ ) equals the zero measure. We call u an unlabeling map.We endow S N with the product topology. For w = { w t } = { ( w it ) } ∈ W ( S N ), weset u path ( w ) t := u ( w t ) = ∞ X i =1 δ w it . (2.3)We call u path ( w ) the unlabeled path of w . Note that u path ( w ) is not necessarily anelement of W ( S ), even if u path ( w ) t ∈ S for all t ; see [22, Remark 3.10], for example.Let S s , i be the subset of S consisting of an infinite number of particles with nomultiple points. By definition, S s , i = S s ∩ S i , where S s and S i are given by S s = { s ∈ S ; s ( { x } ) ≤ x ∈ S } , S i = { s ∈ S ; s ( S ) = ∞} . (2.4)A measurable map l : S s , i → S N is called a label on S s , i if u ◦ l ( s ) = s for all s ∈ S s , i .Let W ( S s ) and W ( S s , i ) be the sets consisting of S s - and S s , i -valued continuouspaths on [0 , ∞ ). Each w ∈ W ( S s ) can be written as w t = P i δ w it , where w i is an S -valued continuous path defined on an interval I i of the form [0 , b i ) or ( a i , b i ), where0 ≤ a i < b i ≤ ∞ . Taking maximal intervals of this form, we can choose [0 , b i ) and( a i , b i ) uniquely up to labeling. We remark that lim t ↓ a i | w it | = ∞ and lim t ↑ b i | w it | = ∞ Infinite-dimensional stochastic differential equations and tail σ -fields II for b i < ∞ for all i . We call w i a tagged path of w and I i the defining interval of w i .Let W NE ( S s , i ) = { w ∈ W ( S s , i ) ; I i = [0 , ∞ ) for all i } . (2.5)It is said that the tagged path w i of w does not explode if b i = ∞ , and does not enter if I i = [0 , b i ), where b i is the right end of the defining interval of w i . Thus, W NE ( S s , i )is the set consisting of non-exploding and non-entering paths.We can naturally lift each w = { P i δ w it } t ∈ [0 , ∞ ) ∈ W NE ( S s , i ) to the labeled path w = ( w i ) i ∈ N = { w t } t ∈ [0 , ∞ ) = { ( w it ) i ∈ N } t ∈ [0 , ∞ ) ∈ W ( S N )using a label l = ( l i ) i ∈ N . Indeed, for each w ∈ W NE ( S s , i ), we can construct the labeledprocess w = { ( w it ) i ∈ N } t ∈ [0 , ∞ ) such that w = l ( w ), because each tagged particle cancarry the initial label i by the non-collision and non-explosion properties of w . Wewrite this correspondence as l path ( w ) = ( l i path ( w )) i ∈ N . (2.6)Setting w = ( w i ) i ∈ N = l path ( w ), we have w i = l i path ( w ) by construction. We remarkthat u path ( w ) t = u ( w t ) by (2.3), whereas l path ( w ) t = l ( w t ) in general.For a labeled path w = ( w i ), we set w m ∗ = { w m ∗ t } t ∈ [0 , ∞ ) by w m ∗ t = P i>m δ w it .We call the path w [ m ] = ( l ( w ) , . . . , l m path ( w ) , w m ∗ ) an m -labeled path. Simi-larly, for a labeled path w = ( w i ) ∈ W ( S N ), we set w [ m ] = ( w , . . . , w m , w m ∗ ) . (2.7) Let X = ( X i ) i ∈ N be an S N -valued continuous process. We write X = { X t } t ∈ [0 , ∞ ) and X i = { X it } t ∈ [0 , ∞ ) . For X and i ∈ N , we define the unlabeled processes X = { X t } t ∈ [0 , ∞ ) and X i ♦ = { X i ♦ t } t ∈ [0 , ∞ ) as X t = P i ∈ N δ X it and X i ♦ t = P j ∈ N , j = i δ X jt .Let H and S sde be Borel subsets of S such that H ⊂ S sde ⊂ S i . Let u [1] : S × S → S be such that u [1] (( x, s )) = δ x + s for x ∈ S and s ∈ S . Define S sde ⊂ S N and S [1]sde ⊂ S × S by S sde = u − ( S sde ) , S [1]sde = u − ( S sde ) . (2.8)Let σ : S [1]sde → R d and b : S [1]sde → R d be Borel measurable functions. We consider anISDE of X = ( X i ) i ∈ N starting on l ( H ) with state space S sde such that dX it = σ ( X it , X i ♦ t ) dB it + b ( X it , X i ♦ t ) dt ( i ∈ N ) , (2.9) X ∈ W ( S sde ) , (2.10) X ∈ l ( H ) . (2.11)Here, B = ( B i ) i ∈ N is an R d N -valued Brownian motion, where R d N = ( R d ) N . By defi-nition, { B i } i ∈ N are independent copies of a d -dimensional Brownian motion startingat the origin. osuke Kawamoto, Hirofumi Osada, and Hideki Tanemura σ and b definedonly on a suitable subset S [1]sde of S × S . From (2.10), the process X moves in the set S sde . Equivalently, the unlabeled dynamics X = u path ( X ) move in S sde . Moreover,each tagged particle X i of X = ( X i ) i ∈ N never explodes.By (2.10), X t ∈ S sde for all t ≥ 0, and in particular the initial starting point s in(2.11) is assumed to satisfy s ∈ l ( H ) ⊂ S sde and equivalently u ( s ) ∈ H ⊂ S sde . Wetake H such that (2.9)–(2.11) have a solution for each s ∈ l ( H ).Following [6, Chapter IV] in finite dimensions, we present a set of notions relatedto solutions of ISDEs. In Definition 2.1, we used the terminology “weak solution”instead of “solution” to distinguish it from the strong solution in Definition 2.4. Definition 2.1 (weak solution) . A weak solution of ISDE (2.9) – (2.10) is an S N × R d N -valued continuous stochastic process ( X , B ) defined on a probability space (Ω , F , P ) with a reference family {F t } t ≥ such that (i) – (iv) hold. (i) X = ( X i ) ∞ i =1 is an S sde -valued continuous process. Furthermore, X is adapted to {F t } t ≥ , that is, X t is F t / B t -measurable for each ≤ t < ∞ , where B t = σ [ w s ; 0 ≤ s ≤ t, w ∈ W ( S N )] . (ii) B = ( B i ) ∞ i =1 is an R d N -valued {F t } -Brownian motion with B = . (iii) The families of measurable {F t } t ≥ -adapted processes Φ i and Ψ i defined by Φ i ( t, ω ) = σ ( X it ( ω ) , X i ♦ t ( ω )) , Ψ i ( t, ω ) = b ( X it ( ω ) , X i ♦ t ( ω )) belong to L and L , respectively. Here, L p is the set of all measurable, {F t } t ≥ -adapted processes α such that E [ R T | α ( t, ω ) | p dt ] < ∞ for all T . We can and do takea predictable version of Φ i and Ψ i (see pp. 45–46 in [6]). (iv) With probability one, the process ( X , B ) satisfies, for all t , X it − X i = Z t σ ( X iu , X i ♦ u ) dB iu + Z t b ( X iu , X i ♦ u ) du ( i ∈ N ) . Definition 2.2 (uniqueness in law) . We say that the uniqueness in law of solutionsstarting on l ( H ) for (2.9) – (2.10) holds if, whenever X and X ′ are two solutions whoseinitial distributions coincide, the laws of the processes X and X ′ on the space W ( S N ) coincide. If this uniqueness holds for an initial distribution δ s , then we say that theuniqueness in law of solutions for (2.9) – (2.10) starting at s holds. Definition 2.3 (pathwise uniqueness) . We say that the pathwise uniqueness of solu-tions for (2.9) – (2.10) starting on l ( H ) holds if, whenever X and X ′ are two solutionsdefined on the same probability space (Ω , F , P ) with the same reference family {F t } t ≥ and the same R d N -valued {F t } -Brownian motion B such that X = X ′ ∈ l ( H ) a.s., P ( X t = X ′ t for all t ≥ 0) = 1 . (2.12) We say that the pathwise uniqueness of solutions starting at s of (2.9) – (2.10) holdsif (2.12) holds whenever the above conditions are satisfied and X = X ′ = s a.s. We now define a strong solution in a form that is an analogous to Definition 1.6in [6, 163p.]. Let P ∞ Br be the distribution of an R d N -valued Brownian motion B with B = . Let W ( R d N ) = { w ∈ W ( R d N ) ; w = } . Clearly, P ∞ Br ( W ( R d N )) = 1.Let B t ( P ∞ Br ) be the completion of σ [ w s ; 0 ≤ s ≤ t, w ∈ W ( R d N )] with respect to P ∞ Br . Let B ( P ∞ Br ) be the completion of B ( W ( R d N )) with respect to P ∞ Br .0 Infinite-dimensional stochastic differential equations and tail σ -fields II Definition 2.4 (strong solution starting at s ) . A weak solution X of (2.9) – (2.10) withan R d N -valued {F t } -Brownian motion B defined on (Ω , F , P, {F t } ) is called a strongsolution starting at s if X = s a.s. and if there exists a function F s : W ( R d N ) → W ( S N ) such that F s is B ( P ∞ Br ) / B ( W ( S N )) -measurable and B t ( P ∞ Br ) / B t -measurable foreach t , and F s satisfies X = F s ( B ) a.s.We also call X = F s ( B ) a strong solution starting at s . Additionally, we call F s itselfa strong solution starting at s . Definition 2.5 (a unique strong solution starting at s ) . We say (2.9) – (2.10) has aunique strong solution starting at s if there exists a function F s : W ( R d N ) → W ( S N ) such that, for any weak solution ( ˆ X , ˆ B ) of (2.9) – (2.10) starting at s , it holds that ˆ X = F s ( ˆ B ) a.s.and if, for any R d N -valued {F t } -Brownian motion B defined on (Ω , F , P, {F t } ) with B = , the continuous process X = F s ( B ) is a strong solution of (2.9) – (2.10) startingat s . Also we call F s a unique strong solution starting at s . We next present a variant of the notion of a unique strong solution. Definition 2.6 (a unique strong solution under constraint) . For a condition ( • ) , wesay (2.9) – (2.10) has a unique strong solution starting at s under the constraint ( • ) ifthere exists a function F s : W ( R d N ) → W ( S N ) such that, for any weak solution ( ˆ X , ˆ B ) of (2.9) – (2.10) starting at s satisfying ( • ) , it holds that ˆ X = F s ( ˆ B ) a.s.and if for any R d N -valued {F t } -Brownian motion B defined on (Ω , F , P, {F t } ) with B = the continuous process X = F s ( B ) is a strong solution of (2.9) – (2.10) startingat s satisfying ( • ) . Also we call F s a unique strong solution starting at s under theconstraint ( • ) . Let ( σ ( x, s ) , b ( x, s )) be the coefficients of ISDE (2.9). We set σ m ( y , s ) = ( σ ( y i , y i ♦ + s )) mi =1 , b m ( y , s ) = ( b ( y i , y i ♦ + s )) mi =1 . (2.13)Here, y i ♦ = P mj = i δ y j for y = ( y , . . . , y m ). For a given unlabeled process X = P ∞ i =1 δ X i , we define the functions σ m X : [0 , ∞ ) × S m → R d and b m X : [0 , ∞ ) × S m → R d such that σ m X ( t, ( u, v )) = σ ( u, v + X m ∗ t ) , b m X ( t, ( u, v )) = b ( u, v + X m ∗ t ) , (2.14)where ( u, v ) ∈ S m , v = u ( v ) := P m − i =1 δ v i ∈ S , where v = ( v , . . . , v m − ) ∈ S m − ,and X m ∗ = ∞ X i = m +1 δ X i . osuke Kawamoto, Hirofumi Osada, and Hideki Tanemura σ m X and b m X depend on both the unlabeled path X and the label l ,although we omit l from the notation for clarity. Let S m sde ( t, X ) be the subset of S m such that S m sde ( t, X ) = { s = ( s , . . . , s m ) ∈ S m ; u ( s ) + X m ∗ t ∈ S sde } . Let ( X , B ) be a weak solution of (2.9)–(2.10) defined on (Ω , F , P, {F t } ). Let P s = P ( ·| X = s ) . (2.15)Then ( X , B ) under P s is a weak solution of (2.9)–(2.10) starting at s = l ( s ). Forsuch a weak solution ( X , B ) defined on (Ω , F , P s , {F t } ), we introduce the SDE withrandom environment X = P i ∈ N δ X i describing Y m = ( Y m,i ) mi =1 given by dY m,it = σ m X ( t, ( Y m,it , Y m,i ♦ t )) dB it + b m X ( t, ( Y m,it , Y m,i ♦ t )) dt, (2.16) Y mt ∈ S m sde ( t, X ) for all t, (2.17) Y m = s m . (2.18)Here, we set Y m,i ♦ = ( Y m,j ) mj = i . Moreover, s m = ( s , . . . , s m ) and B m = ( B , . . . , B m )denote the first m components of s = ( s i ) i ∈ N and B = ( B i ) ∞ i =1 , respectively.We set X m = ( X , . . . , X m ) and X m ∗ = ( X i ) ∞ i = m +1 . Definition 2.7. A triplet ( Y m , B m , X m ∗ ) of {F t } -adapted continuous processes de-fined on (Ω , F , P s , {F t } ) is called a weak solution of (2.16) – (2.18) if it satisfies (2.17) – (2.18) and, for all i ∈ N and t ∈ [0 , ∞ ) , Y m,it − Y m,i = Z t σ m X ( u, ( Y m,iu , Y m,i ♦ u )) dB iu + Z t b m X ( u, ( Y m,iu , Y m,i ♦ u )) du. We also call this a weak solution of (2.16) – (2.17) starting at s m . Clearly, ( X m , B m , X m ∗ ) under (Ω , F , P s , {F t } ) is a weak solution of (2.16)–(2.18)for P ◦ X − -a.s. s . ( B m , X m ∗ ) is given a priori as a part of the coefficients of SDE(2.16).We define the notion of strong solutions and a unique strong solution of (2.16)–(2.18). Let o ∈ S and o m = ( o, . . . , o ) ∈ S m . We set X m ◦∗ = ( o m , X m ∗ ) ∈ W ( S N ).By definition, the first m components of X m ◦∗ consist of the constant path o m . Here, o does not have any special meaning; it can be taken as any point in S . Let e P m = P ◦ ( B m , X m ◦∗ ) − . Let W ( R dm ) = { v ∈ W ( R dm ) ; v = } . We set C m = B ( W ( R dm ) × W ( R d N )) e P m . Let B t ( W ( R dm ) × W ( R d N )) = σ [( v s , w s ); 0 ≤ s ≤ t ]. We set C mt = B t ( W ( R dm ) × W ( R d N )) e P m . Let B mt be the σ -field on W ( R dm ) such that B mt = σ [ w s ; 0 ≤ s ≤ t ].We now state the definition of a strong solution.2 Infinite-dimensional stochastic differential equations and tail σ -fields II Definition 2.8. A weak solution ( Y m , B m , X m ∗ ) of (2.16) – (2.18) defined on (Ω , F , P s , {F t } ) is called a strong solution if there exists a function F m s : W ( R dm ) × W ( R d N ) → W ( R dm ) such that F m s is C m -measurable, C mt / B mt -measurable for all t , and satisfies Y m = F m s ( B m , X m ◦∗ ) a.s. P s . (2.19)For simplicity, we write F m s ( B m , X m ∗ ) := F m s ( B m , X m ◦∗ ). Then (2.19) becomes Y m = F m s ( B m , X m ∗ ) a.s. P s . The solution Y m in Definition 2.8 is defined on (Ω , F , P s , {F t } ), where the weaksolution ( X , B ) is defined. The Brownian motion B m in (2.16) is the first m compo-nents of B , and Y m is a function of not only B m , but also X m ∗ . These propertiesare different from those of the conventional strong solutions of SDEs.Note that for any weak solution ( X , B ), we obtain the weak solution ( X m , B m , X m ∗ )of (2.16)–(2.18). We recall the notion of a unique strong solution from [22]. Definition 2.9. The SDE (2.16) – (2.18) is said to have a unique strong solutionfor ( X , B ) under P s if, for any solution ( ˆ Y m , B m , X m ∗ ) of (2.16) – (2.18) defined on (Ω , F , P s , {F t } ) , there exists a function F m s satisfying ˆ Y m = F m s ( B m , X m ∗ ) a.s. andthe conditions in Definition 2.8. The function F m s in Definition 2.8 is called a strong solution of (2.16)–(2.18). TheSDE (2.16)–(2.18) is said to have a unique strong solution F m s defined on (Ω , F , P s , {F t } )if F m s satisfies the condition in Definition 2.9. The function F m s is unique for e P m -a.s.Following [22], we set the following condition:( IFC ) The SDE (2.16)–(2.18) has a unique strong solution F m s ( B m , X m ∗ ) for ( X , B )under P s for P ◦ X − -a.s. s for all m ∈ N . In Section 2.5, we quote results in [22], which use ( IFC ) as one of the main assump-tions. Similarly as Section 2.4, we set ( X , B ) to be a weak solution of (2.9)–(2.10)defined on (Ω , F , P, {F t } ). Let P s be as (2.15). We quote a sufficient condition for( X , B ) under P s to be a unique strong solution from [22].Let T ( S ) = T ∞ r =1 σ [ π cr ] be the tail σ -field on the configuration space S over R d .Here, π cr is the projection π cr : S → S such that π cr ( s ) = s ( ·∩ S cr ), where S r = {| x | < r } .Let µ be a random point field on S . µ is said to be tail trivial if µ ( A ) ∈ { , } for all A ∈ T ( S ). Let W NE ( S s , i ) be as in (2.5), and set X = u ( X ) as before. For X = ( X i ),we setM r,T ( X ) = inf { m ∈ N ; min t ∈ [0 ,T ] | X it | > r for all i ∈ N such that i > m } . (2.20)We make the following assumptions:( TT ) µ is tail trivial.( AC ) P ◦ X − t ≺ µ for all 0 < t < ∞ .( SIN ) P ( X ∈ W NE ( S s , i )) = 1. osuke Kawamoto, Hirofumi Osada, and Hideki Tanemura NBJ ) P (M r,T ( X ) < ∞ ) = 1 for each r, T ∈ N .We define the conditions ( AC ), ( SIN ), and ( NBJ ) for a probability measure b P on W ( R d N ) by replacing X and X by w and w , respectively.We introduce the condition ( MF ) for a family of strong solutions { F s } of (2.9)–(2.10) starting at P ◦ X − -a.s. s .( MF ) P ( F s ( B ) ∈ A ) is B ( S N ) P ◦ X − -measurable in s for any A ∈ B ( W ( S N )).For a family of strong solutions { F s } satisfying ( MF ) we set P { F s } = Z P ( F s ( B ) ∈ · ) P ◦ X − ( d s ) . (2.21)We remark that, if ( X , B ) is a weak solution under P and a unique strong solutionunder P s for P ◦ X − -a.s. s , then ( MF ) is automatically satisfied and P { F s } = P ◦ X − . (2.22)Here, F s denotes the unique strong solution given by ( X , B ) under P s . Indeed, B isa Brownian motion under both P and P s , and for P ◦ X − -a.s. s P ( F s ( B ) ∈ · ) = P s ( F s ( B ) ∈ · ) = P s ( X ∈ · ) . (2.23)Hence we deduce (2.22) from (2.21) and (2.23). Definition 2.10. For a condition ( • ) , we say (2.9) – (2.10) has a family of uniquestrong solutions { F s } starting at s for P ◦ X − -a.s. s under the constraints of ( MF ) and ( • ) if { F s } satisfies ( MF ) and P { F s } satisfies ( • ) . Furthermore, (i) and (ii) aresatisfied. (i) For any weak solution ( ˆ X , ˆ B ) under ˆ P of (2.9) – (2.10) with ˆ P ◦ ˆ X − ≺ P ◦ X − satisfying ( • ) , it holds that, for ˆ P ◦ ˆ X − -a.s. s , ˆ X = F s ( ˆ B ) ˆ P s -a.s. , where ˆ P s = ˆ P ( ·| ˆ X = s ) . (ii) For an arbitrary R d N -valued {F t } -Brownian motion B defined on (Ω , F , P, {F t } ) with B = , the continuous process X = F s ( B ) is a strong solution of (2.9) – (2.10) satisfying ( • ) starting at s for P ◦ X − -a.s. s . We quote two results from [22]. Both show usefulness of the ( IFC ) condition.Proposition 2.2 (1) is used in [10] to prove the identity ( E , D ) = ( E , D ) explained inSection 1. Proposition 2.1 ([22, Theorem 3.1]) . Assume ( TT ) . Assume that (2.9) – (2.10) hasa weak solution ( X , B ) satisfying ( AC ) , ( SIN ) , ( NBJ ) , and ( IFC ) . Then, (2.9) – (2.10) has a family of unique strong solutions { F s } starting at s for P ◦ X − -a.s. s under the constraints of ( MF ) , ( AC ) , ( SIN ) , ( NBJ ) , and ( IFC ) . Proposition 2.2 ([22, Corollary 3.2]) . Under the same assumptions as Proposi-tion 2.1 the following hold. (1) The uniqueness in law of weak solutions of (2.9) – (2.10) holds under the constraintsof ( AC ) , ( SIN ) , ( NBJ ) , and ( IFC ) . (2) The pathwise uniqueness of weak solutions of (2.9) – (2.10) holds under the con-straints of ( AC ) , ( SIN ) , ( NBJ ) , and ( IFC ) . Infinite-dimensional stochastic differential equations and tail σ -fields II Remark 2.1. (1) All determinantal random point fields on continuous spaces are tailtrivial [21, 13, 1]. Suppose that µ is a quasi-Gibbs measure in the sense of Defini-tion 8.1. Then, µ can be decomposed into tail trivial components, and each componentsatisfies ( AC ) , ( SIN ) , ( NBJ ) , and ( IFC ) . We can apply Proposition 2.1 to eachcomponent ( see [22, Theorem 3.2] ) . Thus, ( TT ) is not restrictive. (2) ( AC ) is obvious if µ is an invariant probability measure of X t and X = µ . Allexamples in the present paper satisfy this condition. ( SIN ) and ( NBJ ) are also mildassumptions. We refer to [22, Sections 10, 12] for sufficient conditions. We shall localize the coefficients of SDE (2.16) to deduce the IFC condition. For this,we introduce a set of subsets in S m × S .Let a = { a q } q ∈ N be a sequence of increasing sequences a q = { a q ( R ) } R ∈ N of naturalnumbers such that a q ( R ) < a q ( R + 1) and a q ( R ) < a q +1 ( R ) for all q, R ∈ N . We set K [ a ] = ∞ [ q =1 K [ a q ] , K [ a q ] = { s ∈ S ; s ( S R ) ≤ a q ( R ) for all R ∈ N } . (3.1)By construction, K [ a q ] ⊂ K [ a q +1 ] for all q ∈ N . It is well known that K [ a q ] is acompact set in S for each q ∈ N .We introduce an approximation of S m × S consisting of compact sets. Let S s , i beas in (2.4). By definition, S s , i is the set consisting of infinite configurations with nomultiple points. Let x = ( x , . . . , x m ) ∈ S m , u ( x ) = P mi =1 δ x i , and s = P i δ s i . Weset S [ m ]s , i = { ( x , s ) ∈ S m × S ; u ( x ) + s ∈ S s , i } . We set S mr = { x ∈ S ; | x | ≤ r } m . Let j, k, l = 1 , . . . , m and S mp,r ( s ) = (cid:8) x ∈ S mr ; inf j = k | x j − x k | > − p , inf l,i | x l − s i | > − p (cid:9) ,S mp,r ( s ) = (cid:8) x ∈ S mr ; min j = k | x j − x k | ≥ − p , inf l,i | x l − s i | ≥ − p (cid:9) . Then S mp,r ( s ) is an open set and S mp,r ( s ) is its closure in S m .Let { a + q ( R ) } R ∈ N be such that a + q ( R ) = 1 + a q ( R + 1). We set H [ a ] ◦ p,q,r = (cid:8) ( x , s ) ∈ S [ m ]s , i ; x ∈ S mp,r ( s ) , s ∈ K [ a + q ] (cid:9) , (3.2) H [ a ] p,q,r = (cid:8) ( x , s ) ∈ S [ m ]s , i ; x ∈ S mp,r ( s ) , s ∈ K [ a + q ] (cid:9) . By construction, H [ a ] ◦ p,q,r is an open set and H [ a ] p,q,r is its closure and compact. Let H [ a ] ◦ r = ∞ [ q =1 H [ a ] ◦ q,r , H [ a ] ◦ q,r = ∞ [ p =1 H [ a ] ◦ p,q,r , H [ a ] r = ∞ [ q =1 H [ a ] q,r , H [ a ] q,r = ∞ [ p =1 H [ a ] p,q,r . We set H [ a ] = ∞ [ r =1 H [ a ] ◦ r = ∞ [ r =1 H [ a ] r . (3.3) osuke Kawamoto, Hirofumi Osada, and Hideki Tanemura H [ a ] ◦ p,q,r and other quantities depend on m ∈ N , we omit m from thenotation.We set N = N ∪ N ∪ N , whereN = { r ∈ N } , N = { ( q, r ) ; q, r ∈ N } , N = { ( p, q, r ) ; p, q, r ∈ N } , and for n ∈ N, we define n + 1 ∈ N asn + 1 = ( p + 1 , q, r ) for n = ( p, q, r ) ∈ N ,( q + 1 , r ) for n = ( q, r ) ∈ N , r + 1 for n = r ∈ N .We write H [ a ] n = H [ a ] p,q,r for n = ( p, q, r ) ∈ N , and set H [ a ] n for n = ( q, r ) ∈ N and n = r ∈ N similarly. We set H [ a ] ◦ n analogously. Clearly, for all n ∈ N H [ a ] ◦ n ⊂ H [ a ] n , H [ a ] n ⊂ H [ a ] n+1 . We shall take the limit in n along with the order n n + 1 such thatlim n →∞ := lim r →∞ lim q →∞ lim p →∞ . For n = ( p, q, r ) ∈ N and ( x , s ) , ( y , s ) ∈ S mp,r ( s ), we set ( x , s ) ∼ n ( y , s ) if x and y are in the same connected component of S mp,r ( s ) and s ∈ Π ( H [ a ] n ). Here Π is aprojection Π : S m × S → S given by Π ( x , s ) = s .Let ( X , B ) be a weak solution of ISDE (2.9)–(2.11) defined on (Ω , F , P, {F t } ). For X = ( X i ) i ∈ N , we set the m -labeled process X [ m ] = ( X m , X m ∗ ) such that X m = ( X , . . . , X m ) , X m ∗ t = ∞ X j = m +1 δ X jt . (3.4)Let ς n ( u , v ) be the exit time from H [ a ] ◦ n . By definition, ς n ( u , v ) is a function on the m -labeled path space W ( S m × S ) such that ς n ( u , v ) = inf { t > u , v ) t H [ a ] ◦ n } . (3.5) { B1 } X [ m ] = ( X m , X m ∗ ) does not exit from H [ a ] = S n ∈ N H [ a ] ◦ n , that is, P ( lim n →∞ ς n ( X m , X m ∗ ) = ∞ ) = 1 . (3.6)We extend the domain of u from S m to S m × S such that u ( x , s ) = u ( x ) + s . Let σ m and b m be as in (2.13). Then we make the following assumptions: { B2 } The inclusion u ( H [ a ]) ⊂ S sde holds. Furthermore, for each n ∈ N and T ∈ N ,there exists a function e F n ,T defined on S m × S satisfying for each f ∈ { σ m , b m } andfor P -a.s. | f ( x , X m ∗ t ) − f ( y , X m ∗ t ) | ≤ | x − y | e F n ,T ( x , X m ∗ t ) (3.7)for all 0 ≤ t ≤ T and all x , y ∈ H [ a ] ◦ n such that ( x , X m ∗ t ) ∼ n ( y , X m ∗ t ). { B3 } The coefficient σ m is a constant function and, for each n ∈ N and T ∈ N , E [ Z T H [ a ] ◦ n ( X m ∗ t , X m ∗ t ) (cid:12)(cid:12) e F n ,T ( X m ∗ t , X m ∗ t ) (cid:12)(cid:12) dt ] < ∞ . (3.8)6 Infinite-dimensional stochastic differential equations and tail σ -fields II { B4 } Filtrations satisfy {F ′ t } = {F ′′ t } and, for each n ∈ N and T ∈ N ,sup { (cid:12)(cid:12) e F n ,T ( x , s ) (cid:12)(cid:12) ; ( x , s ) ∈ H [ a ] n } < ∞ . (3.9)The critical step is to prove the pathwise uniqueness of weak solutions to thefinite-dimensional SDE (2.16) of Y m for ( X , B ). Theorem 3.1. Let ( Z m , ˆ B m , ˆ X m ∗ ) and (ˆ Z m , ˆ B m , ˆ X m ∗ ) be weak solutions of (2.16) – (2.18) defined on (Ω ′ , F ′ , P ′ , {F ′ t } ) and (Ω ′ , F ′ , P ′ , {F ′′ t } ) , respectively. Assume that ( X m , B m , X m ∗ ) law = ( Z m , ˆ B m , ˆ X m ∗ ) law = (ˆ Z m , ˆ B m , ˆ X m ∗ ) . (3.10) Let { B1 } and { B2 } hold for m ∈ N . Let either { B3 } or { B4 } hold for m ∈ N . Then, P ′ ( Z m = ˆ Z m ) = 1 . (3.11)Once the pathwise uniqueness of weak solutions has been established, we candeduce the existence of a unique strong solution through an analogy of the Yamada–Watanabe theory. By using the same argument of the proof of [22, Proposition 11.1],Theorem 3.1 yields the next theorem. Theorem 3.2. Assume that { B1 } , { B2 } , and { B3 } hold for all m ∈ N . Then, ( X , B ) satisfies ( IFC ) . Remark 3.1. It is plausible that Theorem 3.2 holds if we substitute { B3 } by (3.9) .An additional element ˆ X m ∗ prevents us from direct usage of the Yamada–Watanabetheory. Clearly, the condition (3.8) is weaker than (3.9) . For l ∈ { }∪ N , let J [ l ] = { j = ( j k,i ) ≤ k ≤ m, ≤ i ≤ d ; j k,i ∈ { }∪ N , P mk =1 P di =1 j k,i = l } . We set ∂ j = Q k,i ( ∂/∂x k,i ) j k,i for j = ( j k,i ) ∈ J [ l ] , where x k = ( x k,i ) di =1 ∈ R d , and( ∂/∂x k,i ) j k,i denotes the identity if j k,i = 0.We assume that there exists some ℓ = ℓ ( m ) ∈ N satisfying the following. { C1 } For each m ∈ N and n ∈ N , there exists a constant c satisfying the following:For µ [ m ] -a.e. ( x , s ) , ( ξ, s ) ∈ H [ a ] ◦ n satisfying ( x , s ) ∼ n ( ξ, s ), there exists a set of points { x , . . . , x k } in S m with ( x , x k ) = ( x , ξ ) such that k − X j =1 | x j − x j +1 | ≤ c | x − ξ | , [ x j , x j +1 ] × { s } ⊂ H [ a ] n ( j = 1 , . . . , k − , (3.12)and that ∂ j σ mj, s ( t ) and ∂ j b mj, s ( t ) are absolutely continuous in t ∈ [0 , 1] for each j ∈ S ℓ − l =0 J m [ l ] . Here ∂ j σ mj, s ( t ) := ( ∂ j σ m )( t x j + (1 − t ) x j +1 , s ) and we set ∂ j b mj, s ( t ) similarly.Furthermore, [ x j , x j +1 ] is the segment connecting x j and x j +1 . { C2 } For each j ∈ J m [ ℓ ] , there exist g j , h j ∈ C ( S ∩ { x = s } ) such that, on H [ a ], ∂ j σ m ( x , s ) = (cid:16) m X j = k g j ( x k − x j ) + X i g j ( x k − s i ) (cid:17) mk =1 ,∂ j b m ( x , s ) = (cid:16) m X j = k h j ( x k − x j ) + X i h j ( x k − s i ) (cid:17) mk =1 , osuke Kawamoto, Hirofumi Osada, and Hideki Tanemura x = ( x , . . . , x m ) ∈ S m , and the constant c (n) is finite for each n ∈ N : c (cid:8) m X k =1 X i | g j ( x k − s i ) | + | h j ( x k − s i ) | ; ( x , s ) ∈ H [ a ] n (cid:9) < ∞ . (3.13)We refer to [22, Lemma 13.1] for a simple sufficient condition for (3.13). Theorem 3.3. Assume that there exists some ℓ = ℓ ( m ) ∈ N satisfying { C1 } – { C2 } for each m ∈ N . Then, { B2 } , (3.8) , and (3.9) hold for each m ∈ N . We shall give sufficient conditions for { B1 } in Section 5 and Section 6. In this section, we prove Theorem 3.1, Theorem 3.2, and Theorem 3.3. Proof of Theorem 3.1 Let ς n be the exit time from the tame set H [ a ] ◦ n defined by(3.5). From (3.10), we see that ς n ( X m , X m ∗ ) law = ς n ( Z m , ˆ X m ∗ ) law = ς n (ˆ Z m , ˆ X m ∗ ) . (4.1)From { B1 } , we can deduce that ς n ( X m , X m ∗ ) > n := min { ς n ( Z m , ˆ X m ∗ ) , ς n (ˆ Z m , ˆ X m ∗ ) } > . (4.2)From (2.13) and (2.14) we rewrite (2.16) as Y mt − Y m = Z t σ m ( Y mu , X m ∗ u ) d ˆ B mu + Z t b m ( Y mu , X m ∗ u ) du. (4.3)Then ( X m , B m , X m ∗ u ) is a solution of (4.3). Hence, we deduce from (3.10) that( Z m , ˆ B m , ˆ X m ∗ ) and (ˆ Z m , ˆ B m , ˆ X m ∗ ) satisfy Z m = ˆ Z m = s m and Z mt − Z m = Z t σ m ( Z mu , ˆ X m ∗ u ) d ˆ B mu + Z t b m ( Z mu , ˆ X m ∗ u ) du, ˆ Z mt − ˆ Z m = Z t σ m (ˆ Z mu , ˆ X m ∗ u ) d ˆ B mu + Z t b m (ˆ Z mu , ˆ X m ∗ u ) du. From these two equations, we have Z mt − ˆ Z mt = Z t σ m ( Z mu , ˆ X m ∗ u ) d ˆ B mu − Z t σ m (ˆ Z mu , ˆ X m ∗ u ) d ˆ B mu (4.4)+ Z t b m ( Z mu , ˆ X m ∗ u ) du − Z t b m (ˆ Z mu , ˆ X m ∗ u ) du. Assume { B3 } . Then, because σ m is constant by assumption, the difference in themartingale terms of Z m and ˆ Z m is canceled out. Hence, we have from (4.4) Z mt − ˆ Z mt = Z t b m ( Z mu , ˆ X m ∗ u ) − b m (ˆ Z mu , ˆ X m ∗ u ) du. (4.5)8 Infinite-dimensional stochastic differential equations and tail σ -fields II From (3.7), we deduce that, for 0 ≤ u ≤ Σ n ∧ T , | b m ( Z mu , ˆ X m ∗ u ) − b m (ˆ Z mu , ˆ X m ∗ u ) | ≤ | Z mu − ˆ Z mu | e F n ,T ( Z mu , ˆ X m ∗ u ) . (4.6)Combining (4.5) and (4.6), the Schwarz inequality gives for each 0 ≤ t ≤ Σ n ∧ T | Z mt − ˆ Z mt | ≤ { Z t | Z mu − ˆ Z mu | e F n ,T ( Z mu , ˆ X m ∗ u ) du } (4.7) ≤ { Z t | Z mu − ˆ Z mu | du }{ Z t e F n ,T ( Z mu , ˆ X m ∗ u ) du }≤ c Z t | Z mu − ˆ Z mu | du. Here, we set c = R Σ n ∧ T e F n ,T ( Z mu , ˆ X m ∗ u ) du . By (3.8), we see that c < ∞ P -a.s.Hence, from (4.7), we can use Gronwall’s lemma to obtain the identity Z mt = ˆ Z mt until ( Z m , ˆ X m ∗ ) or (ˆ Z m , ˆ X m ∗ ) exit from H [ a ] ◦ n . Then, for all 0 ≤ t ≤ Σ n ∧ T ,( Z mt , ˆ B mt , ˆ X m ∗ t ) = (ˆ Z mt , ˆ B mt , ˆ X m ∗ t ) . (4.8)Taking T → ∞ , we see that (4.8) holds for all 0 ≤ t ≤ Σ n . Because 0 < Σ n by (4.2),this coincidence and the definition of Σ n imply that ς n ( Z m , ˆ X m ∗ ) = ς n (ˆ Z m , ˆ X m ∗ ) = Σ n . Combined with (3.6) and (3.10), this yieldslim n →∞ Σ n = ∞ . (4.9)From (4.8) and (4.9), we obtain (3.11).Next, assume { B4 } . Then the two Brownian motions in (4.4) are equipped withthe same increasing families of σ -fields such that {F ′ t } = {F ′′ t } . Hence we obtain Z t σ m ( Z mu , ˆ X m ∗ u ) d ˆ B mu − Z t σ m (ˆ Z mu , ˆ X m ∗ u ) d ˆ B mu = Z t { σ m ( Z mu , ˆ X m ∗ u ) − σ m (ˆ Z mu , ˆ X m ∗ u ) } d ˆ B mu . Then by the martingale inequality, we have E [sup v ≤ t | Z v ∧ Σ n ∧ T { σ m ( Z mu , ˆ X m ∗ u ) − σ m (ˆ Z mu , ˆ X m ∗ u ) } d ˆ B mu | ] ≤ E [ h Z · { σ m ( Z mu , ˆ X m ∗ u ) − σ m (ˆ Z mu , ˆ X m ∗ u ) } d ˆ B mu i t ∧ Σ n ∧ T ]=4 E [ Z t ∧ Σ n ∧ T tr (cid:0) σ m ( Z mu , ˆ X m ∗ u ) − σ m (ˆ Z mu , ˆ X m ∗ u ) (cid:1) t (cid:0) σ m ( Z mu , ˆ X m ∗ u ) − σ m (ˆ Z mu , ˆ X m ∗ u ) (cid:1) du ] . From (3.7) and (3.9), the last line is dominated by c E [ Z t ∧ Σ n ∧ T | Z mu − ˆ Z mu | e F n ,T ( Z mu , ˆ X m ∗ u ) du ] by (3.7) ≤ c E [ Z t ∧ Σ n ∧ T | Z mu − ˆ Z mu | du ] by (3.9) ≤ c E [ Z t ∧ Σ n ∧ T sup u ≤ v | Z mu − ˆ Z mu | dv ] . osuke Kawamoto, Hirofumi Osada, and Hideki Tanemura c and c are constants depending on d , n ∈ N , and T ∈ N . Hence, we obtain E [sup v ≤ t | Z v ∧ Σ n ∧ T { σ m ( Z mu , ˆ X m ∗ u ) − σ m (ˆ Z mu , ˆ X m ∗ u ) } d ˆ B mu | ] (4.10) ≤ c E [ Z t ∧ Σ n ∧ T sup u ≤ v | Z mu − ˆ Z mu | dv ] . By (3.7) and (3.9) there exists a constant c depending on n ∈ N and T ∈ N suchthatsup v ≤ t | Z v ∧ Σ n ∧ T b ( Z mu , ˆ X m ∗ u ) − b (ˆ Z mu , ˆ X m ∗ u ) du | ≤ c Z t ∧ Σ n ∧ T sup u ≤ v | Z mu − ˆ Z mu | dv. (4.11)From (4.4), we have Z mt − ˆ Z mt = Z t { σ m ( Z mu , ˆ X m ∗ u ) − σ m (ˆ Z mu , ˆ X m ∗ u ) } d ˆ B mu (4.12)+ Z t { b m ( Z mu , ˆ X m ∗ u ) − b m (ˆ Z mu , ˆ X m ∗ u ) } du. Let h ( t ) = E [sup u ≤ t ∧ Σ n ∧ T | Z mu − ˆ Z mu | ]. Then, by (4.10), (4.11), and (4.12) we have h ( t ) ≤ c c Z t h ( u ) du. Hence, by Gronwall’s lemma we obtain h ( t ) = 0 for all t . This implies (3.11).Recall that ( X , B ) under P s is a weak solution of (2.9)–(2.10) starting at s . Thus,( X m , B m , X m ∗ ) becomes a weak solution of (2.16)–(2.18). Proof of Theorem 3.2. The proof of Theorem 3.2 is the same as that of [22, Propo-sition 11.1]. We explain the correspondence and omit the details of the proof.In [22, Proposition 11.1], ( IFC ) was deduced from the pathwise uniqueness of aweak solution. The pathwise uniqueness in [22] was given in (11.6) of Lemma 11.2(3) in [22]. In the present paper, we deduce this pathwise uniqueness as (3.11) inTheorem 3.1. The assumptions in Theorem 3.2 are the same as in Theorem 3.1, andthey are used only to derived the conclusion of Theorem 3.1, that is, the pathwiseuniqueness of weak solutions.The assumptions of [22, Proposition 11.1] are different from those of Theorem 3.2.They were used only to guarantee the existence of weak solutions and the pathwiseuniqueness of weak solutions in Lemma 11.2 (3) in [22]. Hence the proof of [22,Proposition 11.1] is still valid for Theorem 3.2. Proof of Theorem 3.3. For simplicity, we prove the case in which m = 1, ℓ = 2, and d = 1. The general case follows from the same argument.Let ( x, s ) , ( ξ, s ) ∈ H [ a ] ◦ n be such that ( x, s ) ∼ n ( ξ, s ). Then, from { C1 } and d = 1,we see [ x, ξ ] × { s } ⊂ H [ a ] n+1 . From the Taylor expansion b ( x, s ) − b ( ξ, s ) = Z xξ Z yξ ∂ b ( z, s ) dzdy + ( x − ξ ) ∂b ( ξ, s ) . (4.13)0 Infinite-dimensional stochastic differential equations and tail σ -fields II Let c { C2 } , we have that | b ( x, s ) − b ( ξ, s ) | ≤ c (cid:12)(cid:12)(cid:12) Z xξ Z yξ dzdy (cid:12)(cid:12)(cid:12) + | x − ξ || ∂b ( ξ, s ) | (4.14) ≤| x − ξ |{ c r + | ∂b ( ξ, s ) |} . Here, in the last line, we used sup {| x − ξ | ; x, ξ ∈ H [ a ] n+1 } ≤ r √ m = 2 r because m = 1. The same inequality holds for σ . Hence, we take e F n ,T ( x, s ) = { c r + | ∂σ ( x, s ) | + | ∂b ( x, s ) |} . (4.15)We then immediately deduce { B2 } from (4.14) and (4.15).By applying the Taylor expansion as above to ∂σ ( x, s ) and ∂b ( x, s ), we obtainsup {| ∂σ ( x, s ) | + | ∂b ( x, s ) | ; ( x, s ) ∈ H [ a ] n } < ∞ . (4.16)Then, (3.9) follows from (4.15) and (4.16). It is clear that (3.8) follows from (3.9). { B1 } in non-symmetriccase Throughout this section, ( X , B ) is a weak solution of (2.9) and (2.10) defined on(Ω , F , P, {F t } ). We write X = ( X i ) i ∈ N and X [ m ] = ( X m , X m ∗ ).The purpose of this section is to present a sufficient condition for { B1 } . Assump-tion { B1 } implies the non-exit of the m -labeled process X [ m ] from H [ a ] given by(3.3). By definition, H [ a ] is intersection of the set of the single configurations S i andthe tame set K [ a ]. In Section 5.1, we prove the non-exit of the unlabeled dynamics X from S i in Proposition 5.1. In Section 5.2, we prove the non-exit from K [ a ] inProposition 5.7. The main results in the present section are Theorems 5.8 and 5.10given in Section 5.3. Recall that S s is the subset of S consisting of configurations with no multiple points.In this subsection, we derive a sufficient condition such that solutions move in thesubset S s . In other words, we pursue the condition under which particles do notcollide with each other. In many examples, the drift coefficient b is of the form b ( x, s ) = β X i ∇ Ψ( x − s i ) , where s = P i δ s i , and Ψ(0) = ∞ . Hence, b ( x, s ) is not well defined if δ x + s S s .Thus, we need some criterion for the non-collision of particles.We set S R = { x ∈ S ; | x | < R } . For 0 ≤ ǫ ≤ R ∈ N , we set S ,εR = { ( x, y ) ∈ S R ; | x − y | > ε } . Let τ ǫR = τ ǫ,i,jR be the exit time of ( X i , X j ) from S ,εR such that τ ǫR = inf { t > 0; ( X it , X jt ) S ,εR } . (5.1) osuke Kawamoto, Hirofumi Osada, and Hideki Tanemura { C3 } For each R, i = j ∈ N , E [ (cid:12)(cid:12) log | X i − X j | (cid:12)(cid:12) ; ( X i , X j ) ∈ S R × S R ] < ∞ . (5.2) { C4 } For each T, R, i = j ∈ N ,sup ≤ t ≤ T sup <ǫ ≤ E h(cid:12)(cid:12)(cid:12) Z t ∧ τ ǫR (cid:16) X iu − X ju | X iu − X ju | , b ( X iu , X i ♦ u ) (cid:17) R d du (cid:12)(cid:12)(cid:12)i < ∞ . (5.3) { C5 } For each 0 ≤ t < ∞ and i = j ∈ N , E [ Z t S R ( X iu )1 S R ( X ju ) | X iu − X ju | du ] < ∞ . (5.4)Let σ = σ ( x, s ) be the coefficient in (2.9). We set a : S [1]sde → R d such that a = σ t σ. (5.5) { UB } a = ( a kl ( x, s )) dk,l =1 is uniformly elliptic with upper bound c : d X k,l =1 a kl ( x, s ) ξ k ξ l ≤ c | ξ | for all ξ ∈ R d , ( x, s ) ∈ S [1]sde . (5.6) Proposition 5.1. Assume that { C3 } – { C5 } and { UB } hold. Then, P ( X t ∈ S s for all ≤ t < ∞ ) = 1 . (5.7) Proof. For (5.7), it is sufficient to prove that, for each pair ( i, j ) such that i = j , P ( X it = X jt for some 0 ≤ t < ∞ ) = 0 . (5.8)We only prove (5.8) for ( i, j ) = (1 , ϕ ∈ C ∞ (( R d ) ) be such that 0 ≤ ϕ ( x, y ) ≤ ϕ ( x, y ) = ϕ ( y, x ), and ϕ ( x, y ) = ( x, y ) ∈ S ,εR x, y ) S R +1 . Applying Itˆo’s formula to − ϕ ( x, y ) log | x − y | with ( X , X ) and noting that ϕ ( x, y ) =1 on the closure of S ,εR , we then have that, for each 0 < ǫ ≤ R ∈ N , − ϕ ( X t ∧ τ ǫR , X t ∧ τ ǫR ) log | X t ∧ τ ǫR − X t ∧ τ ǫR | = − ϕ ( X , X ) log | X − X | (5.9) − X • Z t ∧ τ ǫR (cid:16) X iu − X ju | X iu − X ju | , σ ( X iu , X i ♦ u ) dB iu (cid:17) R d − X • Z t ∧ τ ǫR (cid:16) X iu − X ju | X iu − X ju | , b ( X iu , X i ♦ u ) (cid:17) R d du + X • Z t ∧ τ ǫR (cid:16) a ( X iu , X i ♦ u ) X iu − X ju | X iu − X ju | , X iu − X ju | X iu − X ju | (cid:17) R d du − X • Z t ∧ τ ǫR | X iu − X ju | d X k =1 a kk ( X iu , X i ♦ u ) du. Infinite-dimensional stochastic differential equations and tail σ -fields II Here, the sum P • is taken over ( i, j ) = (1 , , (2 , i, j ) = (1 , 2) in therest of the proof, and estimate the expectation of each term on the right-hand side of(5.9).A direct calculation, together with (5.5), yields E [ | Z t ∧ τ ǫR (cid:16) X u − X u | X u − X u | σ ( X u , X ♦ u ) , dB u (cid:17) R d | ] (5.10)= E [ D Z · (cid:16) X u − X u | X u − X u | σ ( X u , X ♦ u ) , dB u (cid:17) R d E t ∧ τ ǫR ]= E [ Z t ∧ τ ǫR (cid:16) a ( X u , X ♦ u ) X u − X u | X u − X u | , X u − X u | X u − X u | (cid:17) R d du ] . By (5.6) and (5.4), we can see that for each 0 < ǫ ≤ ≤ t < ∞ E [ Z t ∧ τ ǫR (cid:16) a ( X u , X ♦ u ) X u − X u | X u − X u | , X u − X u | X u − X u | (cid:17) R d du ] (5.11) ≤ c E [ Z t ∧ τ ǫR | X u − X u | du ] by (5.6) ≤ c E [ Z t S R ( X u )1 S R ( X u ) | X u − X u | du ] < ∞ by (5.4) . Next, we prove the L -boundedness of each term of (5.9) in 0 ≤ t ≤ T ∧ τ ǫR and0 < ǫ ≤ T, R ∈ N . By (5.2), the first term on the right-hand side of (5.9) isin L . By (5.10) and (5.11), the second term in (5.9) turns to be L -martingale. Thus,these terms are uniformly integrable. By (5.3), the third term on the right-hand sideof (5.9) is L -bounded. From (5.11), we see that the fourth term on the right-handside is L -bounded. From (5.6) and (5.4), the fifth term on the right-hand side are L -bounded.Collecting these, we have that all the terms on the right-hand side are L -bounded.Thus, we deduce that the left-hand side of (5.9) is L -bounded in 0 ≤ t ≤ T ∧ τ ǫR and0 < ǫ ≤ T, R ∈ N , that is,sup ≤ t ≤ T ∧ τ ǫR , <ǫ ≤ E [ (cid:12)(cid:12)(cid:12) − ϕ ( X t ∧ τ ǫR , X t ∧ τ ǫR ) log (cid:12)(cid:12) X t ∧ τ ǫR − X t ∧ τ ǫR (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ] < ∞ . (5.12)We see that τ R = lim ǫ → τ ǫR because { ( x, x ) ∈ S } is a closed set. Then, taking t → T and then ǫ → 0, we have from Fatou’s lemma and (5.12) that for each T, R ∈ N E h(cid:12)(cid:12) − ϕ ( X T ∧ τ R , X T ∧ τ R ) log (cid:12)(cid:12) X T ∧ τ R − X T ∧ τ R (cid:12)(cid:12)(cid:12)(cid:12)i (5.13) ≤ lim inf ǫ → lim t → T E h(cid:12)(cid:12) − ϕ ( X t ∧ τ ǫR , X t ∧ τ ǫR ) log (cid:12)(cid:12) X t ∧ τ ǫR − X t ∧ τ ǫR (cid:12)(cid:12)(cid:12)(cid:12)i < ∞ . Let τ R be the exit time of ( X , X ) from S R . Then we deduce T ∧ τ R = T ∧ τ R a.s. forall T, R ∈ N from (5.13). Hence, τ R = τ R a.s. for all R ∈ N .By assumption, each tagged particle X i of X = ( X i ) i ∈ N does not explode. Hence,lim R →∞ τ R = ∞ a.s. Together with τ R = τ R a.s. for all R ∈ N , this implieslim R →∞ τ R = ∞ . (5.14) osuke Kawamoto, Hirofumi Osada, and Hideki Tanemura τ = inf { t > 0; ( X t , X t ) ∈ { x = y }} be the first hitting time of ( X , X ) tothe set { x = y } ⊂ S . Then τ = lim R →∞ τ R . Hence (5.14) implies τ = ∞ a.s.Therefore, we deduce that X and X do not collide with each other. K [ a ] . Let K [ a ] and K [ a q ] be the sets given by (3.1). Let κ q be the exit time of X from K [ a q ],that is, κ q = inf { t > X t / ∈ K [ a q ] } . We set κ ∞ := lim q →∞ κ q .In Section 5.2, we shall prove non-exit of X from K [ a ] in such a way that P ( κ ∞ = ∞ ) = 1 . (5.15)The strategy of the proof is to reduce the problem to the construction of a specificfunction e χ on S in (5.34) that diverges on K [ a ] c and satisfies E [ | e χ ( X t ∧ κ ∞ ) | ] < ∞ .For Q ∈ N ∪ {∞} let K Q [ a ] = S ∞ q =1 K Q [ a q ], where K Q [ a q ] is such that K Q [ a q ] = { s ∈ S ; s ( S R ) ≤ a q ( R ) for all R ≤ Q } for Q < ∞ , = { s ∈ S ; s ( S R ) ≤ a q ( R ) for all R < ∞} for Q = ∞ . Then K [ a ] = K Q [ a ] and K [ a q ] = K Q [ a q ] for Q = ∞ .Recall that a = { a q } q ∈ N is a sequence of increasing sequences a q = { a q ( R ) } R ∈ N and that a + q = { a q ( R + 1) } ∞ R =1 for a q = { a q ( R ) } ∞ R =1 . Both {K Q [ a q ] } ∞ q =1 and {K Q [ a + q ] } ∞ q =1 are increasing sequences of compact sets in S if and only if Q = ∞ . Inaddition to (3.1), assume that, for all Q ∈ N ∪ {∞} , K Q [ a q ] ⊂ K Q [ a + q ] ⊂ K Q [ a q +1 ] . (5.16)Note that K Q [ a q ] ⊂ K Q [ a + q ] is clear because a q < a + q . Suppose a q ( R ) = C ( q ) R α forsome α > C ( q ) with C ( q ) → ∞ . Then, taking a newsequence from a = { a q } q ∈ N more intermittently, we can easily retake a such that a + q ( R ) < a q +1 ( R ) for all R ∈ N . For such a we obtain K Q [ a + q ] ⊂ K Q [ a q +1 ]. Then(5.16) holds.We set for Q ∈ N ∪ {∞} L Q [ a q ] = K Q [ a + q ] \K Q [ a q ] . (5.17)Then we have from (5.16) L Q [ a q ] ∩ L Q [ a r ] = ∅ for each q = r ∈ N . (5.18)We next generalize ˇ f given by (2.2) to non-local functions f .Let e S be the set of all countable sums of point measures on S including the zeromeasure. Let S = { S ∞ q =0 S q } S S N as before. For a function f defined on e S , thereexists a unique function ˇ f defined on S such that ˇ f | S m is symmetric in s = ( s i ) mi =1 and that ˇ f ( s ) = f ( u ( s )), where m ∈ N ∪ {∞} and S ∞ = S N .By convention, S denotes the empty set and ˇ f | S is a constant. For a function f on S , we define a function f • on e S by taking f • ( s ) = 0 for s ∈ e S \ S . Then we takeˇ f for f as the restriction of ˇ f • on u − ( S ). The relation between ˇ f and ˇ f S R given by(2.2) for a σ [ S R ]-measurable local function f is, if x , . . . , x m ∈ S R and x j S R for j > m , ˇ f ( x , . . . , x m , x m +1 , , . . . ) = ˇ f S R ( x , . . . , x m ) . (5.19)4 Infinite-dimensional stochastic differential equations and tail σ -fields II Let S mR = S R × · · · × S R be the m -product of S R . Let S mR = { s ∈ S ; s ( S R ) = m } for R, m ∈ N . We set maps π R , π cR : S → S such that π R = π S R and π cR = π S cR .For s ∈ S mR , we call x mR ( s ) = ( x iR ( s )) mi =1 ∈ S mR an S mR -coordinate of s if π R ( s ) = P mi =1 δ x iR ( s ) .For a function f : S → R and R, m ∈ N , we define an S mR -representation f mR, s of f using an S mR -coordinate x mR ( s ) of s . Definition 5.1. We call f mR, s : S mR → R an S mR -representation of f if (1) – (4) hold. (1) f mR, s is a permutation invariant function on S mR for each s ∈ S mR . (2) f mR, s (1) = f mR, s (2) if π cR ( s (1)) = π cR ( s (2)) for s (1) , s (2) ∈ S mR . (3) f mR, s ( x mR ( s )) = f ( s ) for s ∈ S mR . (4) f mR, s ( x mR ( s )) = 0 for s / ∈ S mR . By definition, we have a relation among ˇ f , x mR , and f mR, s such thatˇ f ( x mR ( s ) , s ) = f mR, s ( x mR ( s )) for s ∈ S mR . We say that a function f on S is of C k -class if its S mR -representation f mR, s is in C k ( S mR )for each R, m ∈ N and s ∈ S . Let C k ( S ) be the set consisting of the functions of C k -class. We set C ∞ ( S ) = ∩ ∞ k =0 C k ( S ). Note that a function f on S of C k -class isnot necessary continuous on S because we equip S with the vague topology.Let a be given by (5.5) and D a be the carr´e du champ operator such that D a [ f, g ]( s ) = 12 X i ( a ( s i , s i ♦ ) ∂ ˇ f∂s i , ∂ ˇ g∂s i ) R d . (5.20)By (5.19), we easily see that D a [ f, f ] does not depend on the choice of ˇ f or ˇ f S R for a σ [ π R ]-measurable function f .Next, we introduce a family of cut-off functions { χ q,Q } q ∈ N .We take a label l = ( l i ) such that | l i ( s ) | ≤ | l i +1 ( s ) | for all i . We set for Q ∈ N ∪{∞} d q,Q ( s ) = n Q X R =1 X i ∈ J R, s ( a q ) ( R − | l i ( s ) | ) o / , (5.21)where J R, s ( a q ) = { i ; i > a q ( R ) , l i ( s ) ∈ S R } . Let θ ∈ C ∞ ( R ) such that 0 ≤ θ ( t ) ≤ t ∈ R , θ ( t ) = 0 for t ≤ ǫ , and θ ( t ) = 1 for t ≥ − ǫ for a sufficiently small ǫ > 0. Furthermore, we assume that | θ ′ ( t ) | ≤ √ t . Let χ q,Q ( s ) = θ ◦ d q,Q ( s ) . (5.22) Lemma 5.2. (1) For each q ∈ N and Q ∈ N ∪ {∞} , χ q,Q ∈ C ∞ ( S ) . (2) Assume (5.16) . Then, χ q,Q satisfies the following: ≤ χ q,Q ≤ , χ q,Q ( s ) = ( for s ∈ K Q [ a q ]1 for s 6∈ K Q [ a + q ] , (5.23)0 ≤ D a [ χ q,Q , χ q,Q ] ≤ c , D a [ χ q,Q , χ q,Q ] = 0 for s 6∈ L Q [ a q ] . (5.24) Here K Q [ a q ] , K Q [ a + q ] , and L Q [ a q ] are same as in (5.17) , and c is given by (5.6) . osuke Kawamoto, Hirofumi Osada, and Hideki Tanemura Proof. A direct calculation shows χ q,Q ∈ C ∞ ( S ), (5.23), and the equality in (5.24).Clearly, 0 ≤ D a [ χ q,Q , χ q,Q ]( s ). A straightforward calculation shows that by (5.6) D a [ χ q,Q , χ q,Q ]( s ) ≤ c n θ ′ ( d q,Q ( s )) d q,Q ( s ) o Q X R =1 X i ∈ J R, s ( a q ) ( R − | l i ( s ) | ) = c θ ′ ( d q,Q ( s )) ≤ c . Hence, we see that χ q,Q satisfies the inequalities in (5.24).For N ∈ N and Q ∈ N ∪ {∞} , we set e χ NQ = N X q =1 χ q,Q . (5.25)We regard e χ NQ as a coordinate of s from the viewpoint of K Q [ a q ]. Lemma 5.3. (1) If N and Q ∈ N , then e χ NQ is bounded and continuous on S . (2) If N ∈ N and Q ∈ N ∪ {∞} , then e χ NQ ∈ C ∞ ( S ) . Furthermore, the following hold. q − ≤ e χ NQ ( s ) ≤ q for s ∈ K Q [ a + q ] \K Q [ a q ] , q ≤ N, e χ NQ ( s ) = q for s ∈ K Q [ a q +1 ] \K Q [ a + q ] , q ≤ N, e χ NQ ( s ) = N for s ∈ K Q [ a N +1 ] c , (5.26) D a [ e χ NQ , e χ NQ ]( s ) ( ≤ c for s ∈ K Q [ a N +1 ] , = 0 for s ∈ K Q [ a N +1 ] c . (5.27) Proof. (1) is clear by (5.21), (5.22), and (5.25). (5.26) follows from (5.16), (5.23), and(5.25). The equality in (5.27) follows from (5.26).We finally prove the inequality in (5.27). By (5.25) D a [ e χ NQ , e χ NQ ]( s ) = D a [ N X q =1 χ q,Q , N X q =1 χ q,Q ] = N X q,r =1 D a [ χ q,Q , χ r,Q ] . (5.28)From the Schwarz inequality, (5.18), and (5.24), we have for q = r D a [ χ q,Q , χ r,Q ] ≤ D a [ χ q,Q , χ q,Q ] D a [ χ r,Q , χ r,Q ] = 0 . (5.29)From (5.18) and (5.24), we see for s ∈ K Q [ a ] N X q =1 D a [ χ q,Q , χ q,Q ] = N X q =1 L Q [ a q ] D a [ χ q,Q , χ q,Q ] ≤ c . (5.30)From (5.28), (5.29), and (5.30) we obtain the inequality in (5.27). Let ˇ χ NQ ( s ) =ˇ χ NQ (( s , s , . . . )) be the symmetric function on S N such that ˇ χ NQ ( s ) = e χ NQ ( u ( s )). Recallthat X t = ( X it ) i ∈ N and u ( X t ) = P ∞ i =1 δ X it = X t . Hence, we haveˇ χ NQ ( X t ) = e χ NQ ( u ( X t )) = e χ NQ ( X t ) . (5.31)6 Infinite-dimensional stochastic differential equations and tail σ -fields II We regard ˇ χ NQ as a smooth function on S N ∩ { ˇ χ NQ < ∞} . Let ∂ i = ( ∂ i,k ) dk =1 and set ∂ e χ NQ ∂x k ( x, y ) = ( ∂ ,k ˇ χ NQ )( x, y ) , ∂ e χ NQ ∂x k ∂x l ( x, y ) = ( ∂ ,k ∂ ,l ˇ χ NQ )( x, y ) . (5.32)Here x = ( x , . . . , x d ) ∈ R d and y = u ( y ).Assume N, Q < ∞ . Then, e χ NQ is a local function. Hence, applying Itˆo’s formulato X and ˇ χ NQ together with (5.31) and (5.32), we deduce that e χ NQ ( X t ) is a continuoussemimartingale such that e χ NQ ( X t ) = e χ NQ ( X ) + Z t ∞ X i =1 d X k,l =1 ∂ e χ NQ ∂x k ( X iu , X i ♦ u ) σ kl ( X iu , X i ♦ u ) dB i,lu (5.33)+ Z t ∞ X i =1 d X k =1 b k ( X iu , X i ♦ u ) ∂ e χ NQ ∂x k ( X iu , X i ♦ u ) du + 12 Z t ∞ X i =1 d X k,l =1 a kl ( X iu , X i ♦ u ) ∂ e χ NQ ∂x k ∂x l ( X iu , X i ♦ u ) du. Here, σ = ( σ kl ) dk,l =1 , B i = ( B i,k ) dk =1 , b = ( b k ) dk =1 , and a = ( a kl ) dk,l =1 .By construction, for each s , e χ NQ ( s ) is increasing in Q for each N ∈ N ∪ {∞} , andin N for each Q ∈ N ∪ {∞} . Hence we set e χ ( s ) := lim N →∞ lim Q →∞ e χ NQ ( s ) . (5.34)Then we have e χ ( X t ) = lim N →∞ lim Q →∞ e χ NQ ( X t ) . (5.35)From (5.25) and Lemma 5.3, we see e χ ( s ) < ∞ if and only if s ∈ K [ a ]. Hence e χ ( X t ) < ∞ if and only if X t ∈ K [ a ]. So our task is to prove e χ ( X t ) < ∞ for all t a.s. Lemma 5.4. Assume (5.16) . Assume that ∞ X q =1 q P ( X ∈ K [ a q ] c ) < ∞ . (5.36) Then e χ ( X ) < ∞ a.s. and E [ e χ ( X ) ] < ∞ . (5.37) Proof. From (5.36), we see P ( X ∈ ∩ ∞ q =1 {K [ a q ] c } ) = 0. Then P ( X ∈ K [ a ] c ) = 0.Combining this with (5.26), (5.34), and (5.36), we obtain E [ e χ ( X ) ] = E [1 K [ a ] ( X ) (cid:12)(cid:12)e χ ( X ) (cid:12)(cid:12) ] = lim N →∞ lim Q →∞ E [1 K [ a ] ( X ) (cid:12)(cid:12) e χ NQ ( X ) (cid:12)(cid:12) ] ≤ ∞ X q =1 q P ( e χ NQ ( X ) ∈ K [ a q +1 ] \K [ a q ]) ≤ ∞ X q =1 q P ( e χ NQ ( X ) ∈ K [ a q ] c ) < ∞ . This yields (5.37). The first claim is clear from (5.37). osuke Kawamoto, Hirofumi Osada, and Hideki Tanemura Lemma 5.5. Assume (5.16) . Assume e χ ( X ) < ∞ . Assume that for each t lim N →∞ lim Q →∞ Z t K [ a ] ( X u ) ∞ X i =1 d X k =1 b k ( X iu , X i ♦ u ) ∂ e χ NQ ∂x k ( X iu , X i ♦ u ) du (5.38)= Z t K [ a ] ( X u ) ∞ X i =1 d X k =1 b k ( X iu , X i ♦ u ) ∂ e χ∂x k ( X iu , X i ♦ u ) du a.s. , lim N →∞ lim Q →∞ Z t K [ a ] ( X u ) ∞ X i =1 d X k,l =1 a kl ( X iu , X i ♦ u ) ∂ e χ NQ ∂x k ∂x l ( X iu , X i ♦ u ) du = 12 Z t K [ a ] ( X u ) ∞ X i =1 d X k,l =1 a kl ( X iu , X i ♦ u ) ∂ e χ∂x k ∂x l ( X iu , X i ♦ u ) du a.s. , and that the right-hand sides of the equations in (5.38) are continuous processes andfinite for all t. Then e χ ( X t ) is finite for all t and a continuous semimartingale suchthat e χ ( X t ) = e χ ( X ) + Z t K [ a ] ( X u ) ∞ X i =1 d X k,l =1 ∂ e χ∂x k ( X iu , X i ♦ u ) σ kl ( X iu , X i ♦ u ) dB i,lu (5.39)+ Z t K [ a ] ( X u ) ∞ X i =1 d X k =1 b k ( X iu , X i ♦ u ) ∂ e χ∂x k ( X iu , X i ♦ u ) du + 12 Z t K [ a ] ( X u ) ∞ X i =1 d X k,l =1 a kl ( X iu , X i ♦ u ) ∂ e χ∂x k ∂x l ( X iu , X i ♦ u ) du. Proof. From (5.5) and (5.27), we see that for each t < ∞ E [ (cid:12)(cid:12)(cid:12) Z t ∞ X i =1 d X k,l =1 ∂ e χ NQ ∂x k ( X iu , X i ♦ u ) σ kl ( X iu , X i ♦ u ) dB i,lu (cid:12)(cid:12)(cid:12) ] (5.40)= E [ Z t ∞ X i =1 d X k,l =1 a kl ( X iu , X i ♦ u ) ∂ e χ NQ ∂x k ( X iu , X i ♦ u ) ∂ e χ NQ ∂x l ( X iu , X i ♦ u ) du ]= 2 E [ Z t D a [ e χ NQ , e χ NQ ]( X u ) du ]= 2 E [ Z t K [ a ] ( X u ) D a [ e χ NQ , e χ NQ ]( X u ) du ] ≤ c t. By (5.21), (5.22), and (5.35), we easily see D a [ e χ NQ , e χ NQ ]( s ) are increasing in Q for each N and also D a [ e χ N ∞ , e χ N ∞ ]( s ) are increasing in N . Furthermore,1 K [ a ] ( s ) D a [ e χ, e χ ]( s ) = 1 K [ a ] ( s ) lim N →∞ lim Q →∞ D a [ e χ NQ , e χ NQ ]( s ) (5.41)= lim N →∞ lim Q →∞ D a [ e χ NQ , e χ NQ ]( s ) ≤ c Infinite-dimensional stochastic differential equations and tail σ -fields II and 1 K [ a ] ( s ) D a [ e χ − e χ NQ , e χ − e χ NQ ]( s ) → s along with the limit in Q and N asabove.From (5.20) and (5.41), we deduce that the second term of the right-hand side of(5.39) is a continuous L -martingale and is the limit of the third term of (5.33) in thespace of the continuous L -martingales on (Ω , F , P, {F t } ).By (5.21), (5.22), (5.23), and (5.25), we see for ( x, s ) ∈ R d × S such that δ x + s ∈K [ a ] c ∂ e χ NQ ∂x k ( x, s ) = ∂ e χ NQ ∂x k ∂x l ( x, s ) = 0 . (5.42)Take Q → ∞ and then N → ∞ in (5.33). Then from (5.35), (5.38), and (5.40)–(5.42), we obtain (5.39). Each term of the right-hand side of (5.39) is finite andcontinuous in t by assumption and the argument as above. Hence, e χ ( X t ) < ∞ for all t and e χ ( X t ) is a continuous semimartingale satisfying (5.39). Lemma 5.6. Let κ q be the exit time of X from K [ a q ] . For each t ≥ , sup q ∈ N E [ (cid:12)(cid:12)(cid:12) Z t ∧ κ q ∞ X i =1 d X k,l =1 ∂ e χ∂x k ( X iu , X i ♦ u ) σ kl ( X iu , X i ♦ u ) dB i,lu (cid:12)(cid:12)(cid:12) ] < ∞ . (5.43) Proof. We deduce (5.43) from (5.40) easily. Proposition 5.7. Assume (5.16) , (5.37) , and (5.38) . Assume that sup q ∈ N (cid:12)(cid:12)(cid:12) E [ Z t ∧ κ q ∞ X i =1 d X k =1 b k ( X iu , X i ♦ u ) ∂ e χ∂x k ( X iu , X i ♦ u ) du ] (cid:12)(cid:12)(cid:12) < ∞ , (5.44)sup q ∈ N (cid:12)(cid:12)(cid:12) E [ Z t ∧ κ q ∞ X i =1 d X k,l =1 a kl ( X iu , X i ♦ u ) ∂ e χ∂x k ∂x l ( X iu , X i ♦ u ) du ] (cid:12)(cid:12)(cid:12) < ∞ . Then, we obtain (5.15) . Remark 5.1. Let ι = inf { t > X t / ∈ K [ a ] } . Clearly, κ ∞ ≤ ι . From (5.39) , we have P ( ι = ∞ ) = 1 . We note that P ( ι = ∞ ) = 1 does not imply (5.15) .Proof. Note that e χ NQ are non-negative and continuous for all N, Q ∈ N . Then, by(5.35), the monotone convergence theorem (MCT), and Fatou’s lemma, we obtain foreach t E [ e χ ( X t ∧ κ ∞ )] = lim N →∞ lim Q →∞ E [ e χ NQ ( X t ∧ κ ∞ )] by MCT (5.45) ≤ lim N →∞ lim Q →∞ lim inf q →∞ E [ e χ NQ ( X t ∧ κ q )] by Fatou’s lemma ≤ lim N →∞ lim Q →∞ lim inf q →∞ E [ e χ ( X t ∧ κ q )] by e χ NQ ≤ e χ = lim inf q →∞ E [ e χ ( X t ∧ κ q )] . osuke Kawamoto, Hirofumi Osada, and Hideki Tanemura e χ ( X ) < ∞ a.s. By assumption, (5.16) and (5.38) hold. Then theassumptions of Lemma 5.5 are fulfilled. Hence we obtain (5.39). From (5.39) we see e χ ( X t ∧ κ q ) = e χ ( X ) + Z t ∧ κ q ∞ X i =1 d X k,l =1 ∂ e χ∂x k ( X iu , X i ♦ u ) σ kl ( X iu , X i ♦ u ) dB i,lu (5.46)+ Z t ∧ κ q ∞ X i =1 d X k =1 b k ( X iu , X i ♦ u ) ∂ e χ∂x k ( X iu , X i ♦ u ) du + 12 Z t ∧ κ q ∞ X i =1 d X k,l =1 a kl ( X iu , X i ♦ u ) ∂ e χ∂x k ∂x l ( X iu , X i ♦ u ) du. Taking the expectation for each term in (5.46) and applying (5.37), (5.43), and (5.44)to the right-hand side of (5.46), we deducesup q ∈ N E [ e χ ( X t ∧ κ q )] < ∞ for each t. (5.47)From (5.45) and (5.47), we obtain E [ e χ ( X t ∧ κ ∞ )] < ∞ for each t. (5.48)From (5.48), we see e χ ( X t ∧ κ ∞ ) < ∞ a.s. for each t .From Lemma 5.5, { e χ ( X t ) } is a continuous process. From (5.25), we see e χ ( s ) = ∞ for s / ∈ K [ a ]. Then e χ ( X κ ∞ ) = ∞ a.s. if κ ∞ < ∞ .Combining e χ ( X t ∧ κ ∞ ) < ∞ a.s. for each t and e χ ( X κ ∞ ) = ∞ a.s. for κ ∞ < ∞ , wededuce P ( κ ∞ ≤ t ) = 0 for each 0 ≤ t < ∞ . We therefore obtain P ( κ ∞ < ∞ ) = 0. { B1 } . Theorems 5.8 and 5.10 We now present a sufficient condition of { B1 } for non-symmetric stochastic dynamics.We shall apply Theorem 5.10 to Example 7.8. Theorem 5.8. Assume that { UB } , { C3 } – { C5 } , (5.16) , (5.37) , (5.38) , and (5.44) hold. Then ( X , B ) satisfies { B1 } for each m ∈ N .Proof. { B1 } for m = 0 follows immediately from Proposition 5.1 and Proposition 5.7.Each tagged particle X i has the non-collision and non explosion properties. Then l path is well defined and { B1 } for each m ≥ m = 0. Corollary 5.9. Assume that { UB } , { C3 } – { C5 } , (5.16) , (5.36) , (5.38) , and (5.44) hold. Then ( X , B ) satisfies { B1 } for each m ∈ N .Proof. Corollary 5.9 follows from Lemma 5.4 and Theorem 5.8. Theorem 5.10. Assume that { UB } , { C3 } – { C5 } , (5.16) , and (5.36) . Furthermore,assume λ : law = X is an invariant probability measure of X and Z S ∞ X i =1 d X k =1 (cid:12)(cid:12)(cid:12) b k ( s i , s i ♦ ) ∂ e χ∂x k ( s i , s i ♦ ) (cid:12)(cid:12)(cid:12) dλ < ∞ , (5.49) Z S ∞ X i =1 d X k,l =1 (cid:12)(cid:12)(cid:12) a kl ( s i , s i ♦ ) ∂ e χ∂x k ∂x l ( s i , s i ♦ ) (cid:12)(cid:12)(cid:12) dλ < ∞ . Infinite-dimensional stochastic differential equations and tail σ -fields II Then ( X , B ) satisfies { B1 } for each m ∈ N .Proof. For s ∈ K [ a ], as Q → ∞ and then N → ∞ , we have by (5.16) and Lemma 5.3 ∞ X i =1 d X k =1 (cid:12)(cid:12)(cid:12) b k ( s i , s i ♦ ) ∂ e χ NQ ∂x k ( s i , s i ♦ ) (cid:12)(cid:12)(cid:12) ↑ ∞ X i =1 d X k =1 (cid:12)(cid:12)(cid:12) b k ( s i , s i ♦ ) ∂ e χ∂x k ( s i , s i ♦ ) (cid:12)(cid:12)(cid:12) , (5.50) ∞ X i =1 d X k,l =1 (cid:12)(cid:12)(cid:12) a kl ( s i , s i ♦ ) ∂ e χ NQ ∂x k ∂x l ( s i , s i ♦ ) (cid:12)(cid:12)(cid:12) ↑ ∞ X i =1 d X k,l =1 (cid:12)(cid:12)(cid:12) a kl ( s i , s i ♦ ) ∂ e χ∂x k ∂x l ( s i , s i ♦ ) (cid:12)(cid:12)(cid:12) . Then we deduce (5.38) and (5.44) from (5.49), (5.50), the monotone convergencetheorem, and the assumption that λ is an invariant probability measure of X . Hencewe obtain { B1 } from Corollary 5.9.We remark that (5.49) can be rewritten as Z S × S d X k =1 (cid:12)(cid:12)(cid:12) b k ( x, s ) ∂ e χ∂x k ( x, s ) (cid:12)(cid:12)(cid:12) dλ [1] < ∞ , (5.51) Z S × S d X k,l =1 (cid:12)(cid:12)(cid:12) a kl ( x, s ) ∂ e χ∂x k ∂x l ( x, s ) (cid:12)(cid:12)(cid:12) dλ [1] < ∞ . { B1 } in the symmetriccase Let λ be a random point field such that λ ( S sde ) = 1 and let l : S s , i → S N be a label, asbefore. We shall consider the ISDE (2.9)–(2.10) with the initial distribution λ ◦ l − .Let { Q s } be a family of probability measures on (Ω , F , {F t } ) such that ( X , B )defined on (Ω , F , Q s , {F t } ) is a weak solution of (2.9)–(2.10) starting at s = l ( s ) for λ -a.s. s . We assume { Q s } is a measurable family in the following sense. { MF } Q s ( A ) is B ( S ) λ -measurable in s for each A ∈ F .We remark that { MF } is a counterpart of ( MF ) in Section 2.5. Indeed, λ and Q s correspond to P ◦ X − and P ( F s ( B ) ∈ · ) with s = l ( s ), respectively.For a family of probability measures { Q s } satisfying { MF } , we set Q λ = Z S Q s dλ. Then, ( X , B ) under Q λ is a solution of (2.9)–(2.10) with the initial distribution λ ◦ l − .For m ∈ { } ∪ N , we denote by X [ m ] = ( X m , X m ∗ ) the m -labeled process given by(3.4), where X [0] = X . Let Q [ m ] x , s be the distribution of X [ m ] under Q u ( x )+ s . Then, Z W ( S m × S ) f ( w [ m ] t ) d Q [ m ] x , s = Z f ( X [ m ] t ) d Q u ( x )+ s . (6.1)Let λ [0] = λ . Let λ [ m ] be the m -Campbell measure of λ for m ∈ N . We set for m ∈ { } ∪ N Q [ m ] λ = Z S m × S Q [ m ] x , s dλ [ m ] . (6.2) osuke Kawamoto, Hirofumi Osada, and Hideki Tanemura Q [0] λ = Q λ ◦ X − . We make assumptions. { BX } σ [ B s ; s ≤ t ] ⊂ σ [ X s ; s ≤ t ] for all t under Q λ . { S λ } For each m ∈ { } ∪ N , the m -labeled process X [ m ] under (Ω , F , { Q u ( x )+ s } , {F t } )gives a symmetric, Markovian semi-group T [ m ] t on L ( S m × S , λ [ m ] ) defined by T [ m ] t f ( x , s ) = Z W ( S m × S ) f ( w [ m ] t ) d Q [ m ] x , s . (6.3)Furthermore, λ [ m ] is an invariant measure of T [ m ] t . { D } Let ρ λ is the two-point correlation function of λ . Then, for each R ∈ N , Z S R × S R | x − y | ρ λ ( x, y ) dxdy < ∞ . Theorem 6.1. Assume that { UB } , { MF } , { BX } , { S λ } , and { D } hold. Assume R S | e χ | dλ < ∞ . Then, ( X , B ) under Q λ satisfies { B1 } for each m ∈ { } ∪ N . Remark 6.1. Under { S λ } , a symmetric Dirichlet form associated with the solution ( X , B ) exists through the L -symmetric semi-groups T [ m ] t . However, the Dirichletform is not necessarily quasi-regular. Hence, we can not apply the Dirichlet formtechnique, including the concept of capacity, directly to the solution. We use the factthat ( X , B ) is a solution of (2.9) and the existence of the associated L -symmetricsemi-groups instead. Proposition 6.2. Under the same assumptions as for Theorem 6.1, (5.7) with P = Q λ holds.Proof. We set τ = inf { t > 0; ( X t , X t ) ∈ { x = y }} . Without loss of generality, it issufficient for (5.7) to prove that Q λ ( τ = ∞ ) = 0 . (6.4)Let g ( t ) = − log | t | for t = 0 and g (0) = ∞ . For 0 < ǫ < g ǫ ∈ C ∞ ( R ) suchthat g ǫ ( t ) = g ǫ ( | t | ) for t ∈ R , 0 ≤ g ǫ ( t ) ≤ g ( t ) for | t | ≤ ǫ , and g ǫ ( t ) = g ( t ) for | t | ≥ ǫ .Let G and G ǫ be the functions on S × S such that G ( x , x , s ) = g ( | x − x | ) , G ǫ ( x , x , s ) = g ǫ ( | x − x | ) . (6.5)Then we have (9.4) with m = 2 for G and G ǫ from { UB } , { S λ } and { D } . Weeasily see that { G ǫ ( X [2] t ) } are continuous semimartingales. Applying Lemma 9.2 to { G ǫ ( X [2] t ) } with Q [2] λ , we have G ǫ ( X [2] t ) − G ǫ ( X [2]0 ) = 12 { M [ G ǫ ] t + M [ G ǫ ] T − t ( r T ) − M [ G ǫ ] T ( r T ) } . (6.6)Here, r T : C ([0 , T ]; S × S ) → C ([0 , T ]; S × S ) such that r T ( w [2] )( t ) = w [2] ( T − t ),where w [2] = { w [2] ( t ) } . Furthermore, M [ G ǫ ] is a continuous local martingale additivefunctional of X [2] , where X [2] = ( X , X , P ∞ i =3 δ X i ). By construction, M [ G ǫ ] is givenby M [ G ǫ ] t ( X [2] u ) = X i =1 Z t (cid:0) ∂ i G ǫ ( X [2] ) , σ ( X iu , X i ♦ u ) dB iu (cid:1) R d , Infinite-dimensional stochastic differential equations and tail σ -fields II where X i ♦ = P j = i δ X j . Then, by a straightforward calculation and a = σ t σ , h M [ G ǫ ] ( X [2] ) i t = X i =1 Z t (cid:0) a ( X iu , X i ♦ u ) ∂ i G ǫ ( X [2] u ) , ∂ i G ǫ ( X [2] u ) (cid:1) R d du. (6.7)Note that Q [2] λ is not a probability measure. By abuse of notation, E [2] λ [ · ] denotesthe integral with respect to the measure Q [2] λ and we write X [2]0 law = λ [2] because theimage measure of X [2]0 coincides with λ [2] in the rest of the proof.From { D } , X [2]0 law = λ [2] , and | G ǫ | ≤ | G | , we have E [2] λ [ 1 | X − X | ; X , X ∈ S R ] < ∞ , (6.8)sup <ǫ< E [2] λ [ | G ǫ ( X [2]0 ) | ; X , X ∈ S R ] ≤ E [2] λ [ | G ( X [2]0 ) | ; X , X ∈ S R ] < ∞ . (6.9)Let τ ǫR = τ ǫ, , R in (5.1). From (6.6), we see for 0 ≤ t ≤ TE [2] λ [ (cid:12)(cid:12) G ǫ ( X [2] t ∧ τ ǫR ) − G ǫ ( X [2]0 ) (cid:12)(cid:12) ] (6.10)= 14 E [2] λ [ (cid:12)(cid:12) M [ G ǫ ] t ∧ τ ǫR ( X [2] ) + M [ G ǫ ] T − t ∧ τ ǫR ( r T ( X [2] )) − M [ G ǫ ] T ( r T ( X [2] )) (cid:12)(cid:12) ] ≤ n E [2] λ [ M [ G ǫ ] t ∧ τ ǫR ( X [2] ) ] + E [2] λ [ (cid:12)(cid:12) M [ G ǫ ] T − t ∧ τ ǫR ( r T ( X [2] )) − M [ G ǫ ] T ( r T ( X [2] )) (cid:12)(cid:12) ] o ≤ n E [2] λ [ M [ G ǫ ] T ∧ τ ǫR ( X [2] ) ] + E [2] λ [ (cid:12)(cid:12) M [ G ǫ ] T − T ∧ τ ǫR ( r T ( X [2] )) − M [ G ǫ ] T ( r T ( X [2] )) (cid:12)(cid:12) ] o = E [2] λ [ M [ G ǫ ] T ∧ τ ǫR ( X [2] ) ] by { S λ } . From (6.5), (6.7), { UB } , and { S λ } , we deduce that E [2] λ [ M [ G ǫ ] T ∧ τ ǫR ( X [2] ) ] = E [2] λ [ h M [ G ǫ ] ( X [2] ) i T ∧ τ ǫR ] (6.11) ≤ c E [2] λ [ Z T ∧ τ ǫR | X u − X u | du ] ≤ c T E [2] λ [ 1 | X − X | ; X , X ∈ S R ] . Putting (6.8)–(6.11) together, we deducesup <ǫ< E [2] λ [ (cid:12)(cid:12)(cid:12) G ǫ ( X [2] T ∧ τ ǫR ) (cid:12)(cid:12)(cid:12) ; X , X ∈ S R ] (6.12)= sup <ǫ< E [2] λ [ (cid:12)(cid:12)(cid:12) G ǫ ( X [2] T ∧ τ ǫR ) − G ǫ ( X [2]0 ) + G ǫ ( X [2]0 ) (cid:12)(cid:12)(cid:12) ; X , X ∈ S R ] ≤ n sup <ǫ< E [2] λ [ (cid:12)(cid:12)(cid:12) G ǫ ( X [2] T ∧ τ ǫR ) − G ǫ ( X [2]0 ) (cid:12)(cid:12)(cid:12) ] + sup <ǫ< E [2] λ [ (cid:12)(cid:12)(cid:12) G ǫ ( X [2]0 ) (cid:12)(cid:12)(cid:12) ; X , X ∈ S R ] o < ∞ . Let τ R = lim ǫ → τ ǫR . Then, from Fatou’s lemma and (6.12), we obtain E [2] λ [ | G ( X [2] T ∧ τ R ) | ; X , X ∈ S R ] ≤ lim inf ǫ → E [2] λ [ | G ( X [2] T ∧ τ ǫR ) | ; X , X ∈ S R ]= lim inf ǫ → E [2] λ [ | G ǫ ( X [2] T ∧ τ ǫR ) | ; X , X ∈ S R ] < ∞ . osuke Kawamoto, Hirofumi Osada, and Hideki Tanemura R, T ∈ N S R ( X , X ) | G ( X [2] T ∧ τ R ) | < ∞ Q [2] λ -a.e . (6.13)From (6.5) and (6.13), we see T < τ R holds Q [2] λ -a.e. for all T ∈ N for each R ∈ N .From this combined with (6.1) and (6.2), we see T < τ R holds Q λ -a.s. for all T ∈ N and for each R ∈ N . Hence, Q λ ( τ R < ∞ ) = 0 for each R ∈ N . This implies that Q λ ( τ < ∞ ) = 0, because each tagged particle does not explode. Hence, we obtain(6.4). Proposition 6.3. Make the same assumptions as for Theorem 6.1. Assume (5.16) in addition. Then (5.15) holds.Proof. We use the Lyons-Zheng type decomposition in Lemma 9.2. Note that e χ NQ arelocal smooth and e χ NQ and their derivatives are continuous on S . Applying Lemma 9.2to e χ NQ we have under P = Q λ e χ NQ ( X t ) − e χ NQ ( X ) = 12 { M [ e χ NQ ] t + M [ e χ NQ ] T − t ( r T ) − M [ e χ NQ ] T ( r T ) } (6.14)Here, r T : C ([0 , T ]; S ) → C ([0 , T ]; S ) is such that r T ( w )( t ) = w ( T − t ), where w = { w ( t ) } . Furthermore, M [ e χ NQ ] is a continuous local martingale additive functional of X such that M [ e χ NQ ] t ( X ) = Z t ∞ X i =1 (cid:0) ∂ i e χ NQ ( X u ) , σ ( X iu , X i ♦ u ) dB iu (cid:1) R d (6.15)= Z t K [ a ] ( X u ) ∞ X i =1 (cid:0) ∂ i e χ NQ ( X u ) , σ ( X iu , X i ♦ u ) dB iu (cid:1) R d . By (5.40), (5.41), and the convergence 1 K [ a ] ( s ) D a [ e χ − e χ NQ , e χ − e χ NQ ]( s ) → s along with the limit in Q and N in (5.41), we see the martingales in the right-handside of (6.14) converge to martingales such that e χ ( X t ) − e χ ( X ) = 12 { M [ e χ ] t + M [ e χ ] T − t ( r T ) − M [ e χ ] T ( r T ) } , (6.16) M [ e χ ] t ( X ) = Z t K [ a ] ( X u ) ∞ X i =1 (cid:0) ∂ i e χ ( X u ) , σ ( X iu , X i ♦ u ) dB iu (cid:1) R d . (6.17)Taking the expectation of the square of both sides of (6.16), we have E [ | e χ ( X t ∧ κ q ) − e χ ( X ) | ] = 14 E [ | M [ e χ ] t ∧ κ q + M [ e χ ] T − t ∧ κ q ( r T ) − M [ e χ ] T ( r T ) | ] (6.18) ≤ n E [ | M [ e χ ] t ∧ κ q | ] + E [ | M [ e χ ] T − t ∧ κ q ( r T ) − M [ e χ ] T ( r T ) | ] o ≤ E [ | M [ e χ ] T ∧ κ q | ] . The quadratic variation process h M [ e χ ] i of M [ e χ ] ( X ) is calculated using (6.17). Then,we deduce from (5.41) that E [ | M [ e χ ] T ∧ κ q | ] = E [ h M [ e χ ] i T ∧ κ q ] = 2 E [ Z T ∧ κ q K [ a ] ( X u ) D a [ e χ, e χ ]( X u ) du ] ≤ T c . (6.19)4 Infinite-dimensional stochastic differential equations and tail σ -fields II Combining (6.18) with (6.19), we see that for each 0 ≤ t ≤ T < ∞ sup q ∈ N E [ e χ ( X t ∧ κ q ) ] ≤ 12 sup q ∈ N { E [ | e χ ( X t ∧ κ q ) − e χ ( X ) | ] + E [ e χ ( X ) ] } (6.20) ≤ { T c E [ e χ ( X ) ] } < ∞ by Z S | e χ | dλ < ∞ . Let κ ∞ = lim q →∞ κ q as before. Then, similarly as (5.45), we have by (6.20) E [ e χ ( X t ∧ κ ∞ )] ≤ lim inf q →∞ E [ e χ ( X t ∧ κ q )] < ∞ for each 0 ≤ t < ∞ . (6.21)From (6.21), we have for each 0 ≤ t < ∞ e χ ( X t ∧ κ ∞ ) < ∞ a.s . (6.22)From (5.26), we obtain e χ ( X κ ∞ ) = ∞ . (6.23)From (6.22) and (6.23), we deduce Q λ ( t < κ ∞ ) = 1 for each t . This implies (5.15). Proof of Theorem 6.1. Theorem 6.1 follows from Proposition 6.2 and Proposition 6.3. Following [22], we present examples satisfying the assumptions of the main theorems.In all the examples in this section, σ is the unit matrix. In Example 7.1–Example 7.7, b ( x, y ) = d µ ( x, y ), where d µ is the logarithmic derivative of the random point field µ associated with the ISDE.The first three examples are infinite particle systems in one-dimensional space,and the fourth example is in R . These four examples arise from the random matrixtheory and have logarithmic interaction potential.Example 7.5–Example 7.7 are related to Ruelle’s class interaction potentials. Theequilibrium states for these examples are canonical Gibbs measures described by theDobrushin–Lanford–Ruelle (DLR) equation. We consider only non-symmetric solu-tions in Example 7.8. Here, non-symmetric means the associated unlabeled dynamicsare not reversible to the given equilibrium state. We construct such dynamics byadding skew-symmetric drift coefficients. Example 7.1 (sine β random point fields) . Let S = R . We consider dX it = dB it + β r →∞ ∞ X | X it − X jt | Example 7.4 (Ginibre random point field) . Let S = R and β = 2 . We considertwo ISDEs: dX it = dB it + lim r →∞ X | X it − X jt | The next two examples are individual cases of Example 7.5. We present only theinteraction potentials and ISDEs. Example 7.6 (Lennard–Jones 6-12 potentials) . Let d = 3 , β > , and Ψ , ( x ) = {| x | − − | x | − } . The interaction Ψ , is called the Lennard–Jones 6-12 potential. The ISDE is dX it = dB it + β ∞ X j =1 ,j = i { X it − X jt ) | X it − X jt | − X it − X jt ) | X it − X jt | } dt ( i ∈ N ) . Example 7.7 (Riesz potentials) . Let d < a ∈ R , < β , and set Ψ a ( x ) = ( β/a ) | x | − a .The corresponding ISDE is dX it = dB it + β ∞ X j =1 ,j = i X it − X jt | X it − X jt | a +2 dt ( i ∈ N ) . Example 7.8 (Non-symmetric case) . Let Ψ be a Ruelle’s class potential. We assume Ψ ∈ C ( R d ) and d ≥ . Let µ be an associated canonical Gibbs measure. We take λ = µ . We assume that µ has locally bounded m -point correlation functions for all m . Then, the logarithmic derivative of µ is given by d µ ( x, s ) = − β X i ∇ Ψ ( x − s i ) . Let γ be an R d -valued function on R d such that γ ∈ C ( R d ) . Let γ ( x, s ) = β X i γ ( x − s i ) . We consider the ISDE dX it = dB it + 12 { d µ ( X it , X i ♦ t ) + γ ( X it , X i ♦ t ) } dt (7.10)8 Infinite-dimensional stochastic differential equations and tail σ -fields II under the assumption that div γ + γ · d µ = 0 . (7.11) An example of Ψ , µ , and γ satisfying (7.11) is Ψ = 0 , λ = µ is the Poissonrandom point field whose intensity is the Lebesgue measure, and γ = ( γ k ) dk =1 is thederivative of a skew-symmetric potential Γ = (Γ kl ) dk,l =1 such that γ ( x ) = d X k =1 ∂ Γ kl ∂x l ( x ) , Γ kl ( x ) = − Γ lk ( x ) . Here x = ( x , . . . , x d ) ∈ R d . From (7.11) , µ [ m ] is an invariant measure of X [ m ] .To apply Theorem 3.1 and Theorem 3.2, we check { B1 } , { B2 } , and { B3 } . { B2 } and { B3 } follow from Theorem 3.3. Indeed, we can take ℓ = 1 because Ψ and Γ arecompact supports. As for the construction of a weak solution of (7.10) , we can useLemma 8.1. We can take a suitable finite particle approximation µ N because Ψ is ofRuelle’s class and is a compact support. Moreover, γ is also a compact support.To obtain { B1 } , we use Theorem 5.10. So we quickly check the assumptions ofTheorem 5.10. { UB } is obvious. { C3 } is clear because the two-point correlationfunction is bounded. { C4 } and { C5 } follow from boundedness of the two-point corre-lation function, d ≥ , and the assumptions such that Ψ ∈ C ( R d ) and γ ∈ C ( R d ) and that µ [ m ] are invariant measures of X [ m ] . It is not difficult to see that µ satisfies (5.16) , (5.36) , and (5.49) if µ is a translation invariant Poisson random point field.Hence, we obtain { B1 } from Theorem 5.10.It is plausible that one can generalize the example to canonical Gibbs measuresand the long-range case. For this, more work is required and is left to the reader. In this section, we quickly review some previous results. In [17, 18, 19, 20, 8], wepresented weak solutions to ISDEs; [17, 18, 19, 20] were devoted to symmetric cases,whereas [8] considered both non-symmetric and symmetric cases. Hence, togetherwith the results in [22] and the present paper, we obtain unique strong solutions ofISDEs. In Section 8.1, we follow the process for constructing weak solutions in [8]. The resultsare valid for non-symmetric solutions.Let { µ N } be a sequence of random point fields on S such that µ N ( { s ( S ) = N } ) =1. Let l N be a label of µ N and l Nm = ( l N, , . . . , l N,m ), where m ≤ N . We assume thefollowing. { H1 } Each µ N has a correlation function { ρ N,n } with respect to the Lebesgue measuresatisfying, for each r ∈ N ,lim N →∞ ρ N,n ( x ) = ρ n ( x ) uniformly on S nr for all n ∈ N , (8.1)sup N ∈ N sup x ∈ S nr ρ N,n ( x ) ≤ c n n c n for all n ∈ N , (8.2)where 0 < c ( r ) < ∞ and 0 < c ( r ) < n ∈ N . osuke Kawamoto, Hirofumi Osada, and Hideki Tanemura { H2 } For each m ∈ N , lim N →∞ µ N ◦ ( l Nm ) − = µ ◦ ( l m ) − weakly in S m .We take µ N ◦ ( l N ) − as an initial distribution of the labeled finite-particle system,and { H2 } refers to the convergence of the initial distribution of the labeled dynamics.For X N = ( X N,i ) Ni =1 , we set X N,i ♦ t = P Nj = i δ X N,jt , where X N,i ♦ t denotes the zeromeasure for N = 1. Let σ N : S × S → R d and b N : S × S → R d be measurablefunctions. The finite-dimensional SDE of X N = ( X N,i ) Ni =1 is given by dX N,it = σ N ( X N,it , X N,i ♦ t ) dB it + b N ( X N,it , X N,i ♦ t ) dt (1 ≤ i ≤ N ) , (8.3) X N = s . (8.4) { H3 } SDE (8.3) and (8.4) has a weak solution for µ N ◦ ( l N ) − -a.s. s for each N andthis solution neither explodes nor hits the boundary (when ∂S is non-void). { H4 } σ N are bounded and continuous on S × S , and converge uniformly to σ on S r × S for each r ∈ N . Furthermore, a N := σ N t σ N are uniformly elliptic on S r × S for each r ∈ N and ∂∂x a N ( x, s ) are uniformly bounded on S × S .Let X N,mT be the maximal module variable of the first m particles such that X N,mT = max i =1 ,...,m sup t ∈ [0 ,T ] | X N,it | . { I1 } For each T, m ∈ N ,lim a →∞ lim inf N →∞ P µ N ◦ ( l N ) − ( X N,mT ≤ a ) = 1and there exists a constant c = c m, a ) such that, for 0 ≤ t, u ≤ T ,sup N ∈ N m X i =1 E µ N ◦ ( l N ) − [ | X N,it − X N,iu | ; X N,mT ≤ a ] ≤ c | t − u | . Furthermore, M r,T , defined by (2.20), satisfieslim L →∞ lim inf N →∞ P µ N ◦ ( l N ) − (M r,T ( X N ) ≤ L ) = 1 for each r ∈ N . Let µ N, [1] be the one-Campbell measure of µ N . Set c ( r, N ) = µ N, [1] ( S r × S ).Then, by (8.2), sup N c r, N ) < ∞ for each r ∈ N . Without loss of generality, wecan assume that c > r, N . Let µ N, [1] r = µ N, [1] ( · ∩ { S r × S } ). Let ¯ µ N, [1] bethe probability measure defined as¯ µ N, [1] ( · ) = µ N, [1] ( · ∩ { S r × S } ) /c . Let ̟ r,s be the map from S r × S to itself such that ̟ r,s ( x, s ) = ( x, P | x − s i | A random point field µ is called a (Φ , Ψ) -quasi Gibbs measure withinverse temperature β > if its regular conditional probabilities µ mr,ξ = µ ( π r ( x ) ∈ · | π cr ( x ) = π cr ( ξ ) , x ( S r ) = m ) satisfy, for all r, m ∈ N and µ -a.s. ξ , c − e − β H r ( x ) Λ mr ( d x ) ≤ µ mr,ξ ( d x ) ≤ c e − β H r ( x ) Λ mr ( d x ) . Here, c = c r, m, ξ ) is a positive constant depending only on r, m, π cr ( ξ ) . For twomeasures µ, ν on a σ -field F , we write µ ≤ ν if µ ( A ) ≤ ν ( A ) for all A ∈ F . We make the following assumptions. { A1 } µ is a (Φ , Ψ)-quasi Gibbs measure such that there exist upper semi-continuousfunctions ( ˆΦ , ˆΨ) and positive constants c and c satisfying c − 16 ˆΦ( x ) ≤ Φ( x ) ≤ c 16 ˆΦ( x ) , c − 17 ˆΨ( x, y ) ≤ Ψ( x, y ) ≤ c 17 ˆΨ( x, y ) . { A2 } For each r ∈ N , µ satisfies P ∞ m =1 mµ ( S mr ) < ∞ .Let ( E a,µ , D a,µ ◦ ) be the bilinear form on L ( S , µ ) with domain D a,µ ◦ defined by E a,µ ( f, g ) = Z S D a [ f, g ] µ ( d s ) . Here D a is as in (5.20) and D a,µ ◦ = { f ∈ D ◦ ∩ L ( S , µ ) ; E a,µ ( f, f ) < ∞} , where D ◦ is the set of all bounded, local smooth functions on S .A family of probability measures { P s } s ∈ S on ( W ( S ) , B ( W ( S ))) is called a dif-fusion if the canonical process X = { X t } under P s is a continuous process with thestrong Markov property starting at s . Here, X t ( w ) = w t for w = { w t } ∈ W ( S ) bydefinition. X is adapted to {F t } , where F t = ∩ ν F νt and the intersection is taken overall Borel probability measures ν ; F νt is the completion of F + t = ∩ ǫ> B t + ǫ ( S ) withrespect to P ν = R P s ν ( d s ), where B t ( S ) = σ [ w s ; 0 ≤ s ≤ t ]. Furthermore, { P s } s ∈ S is said to be ν -stationary if ν is an invariant probability measure. We say { P s } s ∈ S is ν -reversible if { P s } s ∈ S is ν -symmetric and -stationary. Lemma 8.2 ([16, 19, 22]) . Assume that { A1 } and { A2 } hold. Then, ( E a,µ , D a,µ ◦ ) isclosable on L ( S , µ ) , and its closure ( E a,µ , D a,µ ) is a quasi-regular Dirichlet form on L ( S , µ ) . Moreover, the associated µ -reversible diffusion ( X , { P s } s ∈ H ) exists. We refer to [14] for the definition of quasi-regular Dirichlet forms and relatednotions. We also refer to [4] for details of Dirichlet form theory.Let { P s } s ∈ H be as in Lemma 8.2. Note that µ ( H ) = 1 and set P µ = R H P s µ ( d s ).Let l path be as in (2.6). Let S sde and S sde be as in (2.8). Then we assume thefollowing: { A3 } P µ ( W NE ( S s , i )) = 1 and P µ ◦ l − ( W ( S sde )) = 1. Definition 8.2 ([18]) . An R d -valued function d µ is called the logarithmic derivativeof µ if d µ ∈ L ( S × S , µ [1] ) and, for all ϕ ∈ C ∞ ( S ) ⊗ D ◦ , Z S × S d µ ( x, s ) ϕ ( x, s ) µ [1] ( dxd s ) = − Z S × S ∇ x ϕ ( x, s ) µ [1] ( dxd s ) . Here we write f ∈ L p loc ( S × S , µ [1] ) if f ∈ L p ( S r × S , µ [1] ) for all r ∈ N , and we set ∇ x a ( x, s ) = ( ∂a kl ∂x l ( x, s )) dl =1 , where x = ( x , . . . , x d ) . osuke Kawamoto, Hirofumi Osada, and Hideki Tanemura { A4 } µ has a logarithmic derivative d µ , and the coefficients ( σ, b ) satisfy σ t σ = a, b = 12 ∇ x a + 12 d µ . Lemma 8.3 ([18, Theorem 26]) . Assume that { A1 } – { A4 } hold. Then, there existssome {F t } -Brownian motion B such that ( l path ( X ) , B ) is a weak solution of (2.9) . In Lemma 8.3, (Ω , F , P s , {F t } ) is the filtered space introduced before Lemma 8.2.We can take H in (2.11) uniquely up to capacity zero and S sde in (2.10) as u − ( H ).ISDE (2.9)–(2.11) has a weak solution ( l path ( X ) , B ) defined on (Ω , F , P s , {F t } ) foreach s ∈ H and P µ ( X t H for some 0 ≤ t < ∞ ) = 0 and, in particular, µ ( H ) = 1. Let ( X , B ) be a weak solution of (2.9)–(2.10). We shall derive the Lyons–Zheng typedecomposition of additive functionals of X .Let m ∈ { }∪ N and F ∈ C ( S m × S i ), where S i is given by (2.4) and C ( S m × S i )is the set of functions on S m × S i of C -class. Here, we say that F is of C -class ifˇ F ∈ C ( S N ) in the sense that ˇ F ( s , . . . , s n , s n +1 , . . . ) is C in ( s , . . . , s n ) for fixed( s n +1 , . . . ) for all n ∈ N . The function ˇ F is such that, for x = ( x , . . . , x m ) and s = P ∞ i = m +1 δ s i , F ( x , s ) = ˇ F ( x , . . . , x m , s m +1 , s m +2 , s m +3 , . . . )and for any permutation p on N \{ , . . . , m } ,ˇ F ( x , . . . , x m , s m +1 , s m +2 , s m +3 , . . . ) = ˇ F ( x , . . . , x m , s p ( m +1) , s p ( m +2) , s p ( m +3) , . . . ) . Let w [ m ] be as in (2.7). Let r T : C ([0 , T ]; S m × S i ) → C ([0 , T ]; S m × S i ) be suchthat r T ( w [ m ] )( t ) = w [ m ] ( T − t ) . (9.1)We regard Q [ m ] λ as a measure on C ([0 , T ]; S m × S i ), where Q [ m ] λ is given by (6.2).Indeed, by (2.10) each tagged particle X i of X does not explode under Q λ . Hence,each tagged path of w [ m ] does not explode for Q [ m ] λ -a.e. w [ m ] . Lemma 9.1. Q [ m ] λ = Q [ m ] λ ◦ r − T .Proof. By (6.2), Q [ m ] λ = R S m × S Q [ m ] x , s dλ [ m ] . We deduce from { S λ } that { Q [ m ] x , s } is a λ [ m ] -symmetric Markov process and λ [ m ] is an invariant measure of { Q [ m ] x , s } . Fromthese we conclude Lemma 9.1.Let X [ m ] be the m -labeled process of X , that is, X [ m ] = ( X , . . . , X m , P ∞ i = m +1 δ X i ).Recall that B is a function of X and σ [ B s ; s ≤ t ] ⊂ σ [ X s ; s ≤ t ] for all t under Q λ by { BX } . Then, there exists ˆ B m such that B m = ˆ B m ( X [ m ] ) under Q λ . Clearly, ˆ B m is a dm -dimensional Brownian motion under Q [ m ] x , s for λ [ m ] -a.e. ( x , s ). Here we recall Q [ m ] λ is not necessary a probability measure for m ∈ N . Below, “under Q [ m ] λ ” means“under Q [ m ] x , s for λ [ m ] -a.e. ( x , s )”.4 Infinite-dimensional stochastic differential equations and tail σ -fields II Let w m = ( w , . . . , w m ), where w [ m ] = ( w m , w m ∗ ). Then w m under Q [ m ] λ is aweak solution of (2.16) with Brownian motion ˆ B m = ( ˆ B m,i ) mi =1 and X = w , where w = P ∞ i =1 δ w i as before. The coefficients of (2.16) depends only on X m ∗ = P ∞ i = m +1 δ X i ,so does w m ∗ in the present case. By (2.13), we can rewrite (2.16) as dw it = σ ( w it , w i ♦ t ) d ˆ B m,it + b ( w it , w i ♦ t ) dt for i = 1 , . . . , m. (9.2)Here, for w = ( w i ) ∞ i =1 , we set w i ♦ = { w i ♦ t } t by w i ♦ t = P ∞ j = i, j =1 δ w jt .Let S m = = { x = ( x i ) mi =1 ; x i = x j for all i = j } and F ∈ C ( S m = × S ). Below, wewrite w [ m ] ( t ) = w [ m ] t . Applying Itˆo’s formula to F informally, we see under Q [ m ] λ F ( w [ m ] t ) − F ( w [ m ]0 ) = Z t ∞ X i =1 (cid:0) ∂ i ˇ F ( w u ) , σ ( w iu , w i ♦ u ) d ˆ B m,iu (cid:1) R d + (9.3) Z t ∞ X i =1 (cid:0) b ( w iu , w i ♦ u ) , ∂ i ˇ F ( w u ) (cid:1) R d du + Z t ∞ X i =1 d X k,l =1 a kl ( w iu , w i ♦ u ) ∂ i,k ∂ i,l ˇ F ( w u ) du. The equality (9.3) can be justified if F is a local smooth function, and each term isintegrable. We shall assume F ( w [ m ] t ) is a continuous semimartingale satisfying (9.3). Lemma 9.2. Consider the same assumptions as for Theorem 6.1. Let m ∈ { } ∪ N and F ∈ C ( S m = × S ) . Assume that for Q [ m ] λ -a.e. w [ m ] Z t ∞ X i =1 d X k,l =1 a kl ( w iu , w i ♦ u ) ∂ i,k ˇ F ( w u ) ∂ i,l ˇ F ( w u ) du < ∞ for all t (9.4) and that F ( w [ m ] t ) is a continuous semimartingale under Q [ m ] λ satisfying (9.3) . Then,under Q [ m ] λ , we obtain for ≤ t ≤ TF ( w [ m ] t ) − F ( w [ m ]0 ) = 12 n M t ( w [ m ] ) + (cid:0) M T − t ( r T ( w [ m ] )) − M T ( r T ( w [ m ] )) (cid:1)o . (9.5) Here, M is a continuous local martingale under Q [ m ] λ such that M t = Z t ∞ X i =1 (cid:0) ∂ i ˇ F ( w u ) , σ ( w iu , w i ♦ u ) d ˆ B m,iu (cid:1) R d . (9.6) The quadratic variation of M is given by h M i t ( w [ m ] ) = Z t ∞ X i =1 d X k,l =1 a kl ( w iu , w i ♦ u ) ∂ i,k ˇ F ( w u ) ∂ i,l ˇ F ( w u ) du. (9.7) Furthermore, { M T − t ( r T ( w [ m ] )) − M T ( r T ( w [ m ] )) } is a continuous local martingaleunder Q [ m ] λ with respect to the inverse filtering.Proof. We modify the argument of [4, Theorem 5.7.1] according to the current sit-uation. Note that the weak solution ( X , B ) is not associated with any quasi-regularDirichlet forms; there exists no L -semi-group associated with the labeled process osuke Kawamoto, Hirofumi Osada, and Hideki Tanemura X . We can still use the L -semi-group associated with the m -labeled process X [ m ] (equivalently, w [ m ] under Q [ m ] λ ) given by (6.3) for any m ∈ N .For x = ( x i ) ∞ i =1 , we set x [ m ] = ( x , . . . , x m , P ∞ i = m +1 δ x i ) and x i ♦ = P ∞ j = i, j =1 δ x j .Let G ( x [ m ] ) = ∞ X i =1 (cid:0) b ( x i , x i ♦ ) , ∂ i ˇ F ( x ) (cid:1) R d + ∞ X i =1 d X k,l =1 a kl ( x i , x i ♦ ) ∂ i,k ∂ i,l ˇ F ( x ) . (9.8)Then, from (9.3), (9.6), and (9.8), we have that under Q [ m ] λ for 0 ≤ t ≤ TF ( w [ m ] t ) − F ( w [ m ]0 ) = M t ( w [ m ] ) + Z t G ( w [ m ] u ) du. (9.9)By Lemma 9.1, Q [ m ] λ = Q [ m ] λ ◦ r − T . Hence, M t ( r T ( w [ m ] )) is well-defined for Q [ m ] λ -a.e. w [ m ] . We see then from (9.9) the following. F ( r T ( w [ m ] ) t ) − F ( r T ( w [ m ] ) ) = M t ( r T ( w [ m ] )) + Z t G ( r T ( w [ m ] ) u ) du. (9.10)By the definition of r T , we can rewrite (9.10) as F ( w [ m ] T − t ) − F ( w [ m ] T ) = M t ( r T ( w [ m ] )) + Z T G ( w [ m ] u ) du − Z T − t G ( w [ m ] u ) du. (9.11)Hence, from (9.11), we obviously have M t ( r T ( w [ m ] )) = F ( w [ m ] T − t ) − F ( w [ m ] T ) − Z T G ( w [ m ] u ) du + Z T − t G ( w [ m ] u ) du. (9.12)Take t to be T − t and T in (9.12). Then we have M T − t ( r T ( w [ m ] )) = F ( w [ m ] t ) − F ( w [ m ] T ) − Z T G ( w [ m ] u ) du + Z t G ( w [ m ] u ) du, (9.13) M T ( r T ( w [ m ] )) = F ( w [ m ]0 ) − F ( w [ m ] T ) − Z T G ( w [ m ] u ) du. (9.14)Subtract both sides of (9.14) from those of (9.13). Then using (9.9) we obtain M T − t ( r T ( w [ m ] )) − M T ( r T ( w [ m ] )) = F ( w [ m ] t ) − F ( w [ m ]0 ) + Z t G ( w [ m ] u ) du =2 { F ( w [ m ] t ) − F ( w [ m ]0 ) } − M t ( w [ m ] ) . Hence, we have under Q [ m ] λ for 0 ≤ t ≤ TF ( w [ m ] t ) − F ( w [ m ]0 ) = 12 n M t ( w [ m ] ) + (cid:0) M T − t ( r T ( w [ m ] )) − M T ( r T ( w [ m ] )) (cid:1)o . This completes the proof of (9.5). Equation (9.7) follows immediately from (9.6).The last claim follows from Lemma 9.1 and the definition of r T .6 Infinite-dimensional stochastic differential equations and tail σ -fields II 10 Acknowledgments. H.O. is supported in part by a Grant-in-Aid for Scientific Research (Grant Nos.16K13764, 16H02149, 16H06338, and KI BAN-A, No. 24244010) from the JapanSociety for the Promotion of Science. H.T. is supported in part by a Grant-in-Aid forScientific Research(Scientific Research (B), No. 19H01793) from the Japan Societyfor the Promotion of Science. References [1] Bufetov I.A., Qiu Yanqi, Shamov A.: Kernels of conditional determinantal mea-sures and the proof of the Lyons-Peres conjecture . (to appear in J. Eur. 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