Finite-particle approximations for interacting Brownian particles with logarithmic potentials

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1 c 218 The Mahemaical Sociey of Japan J. Mah. Soc. Japan Vol. 7, o. 3 (218) pp doi: /jmsj/ Finie-paricle approximaions for ineracing Brownian paricles wih logarihmic poenials By Yoske Kawamoo and Hirofmi Osada (Received Jly 23, 216) (Revised ov. 29, 216) Absrac. We prove he convergence of -paricle sysems of Brownian paricles wih logarihmic ineracion poenials ono a sysem described by he infinie-dimensional sochasic differenial eqaion (ISDE). For his proof we presen wo general heorems on he finie-paricle approximaions of ineracing Brownian moions. In he firs general heorem, we presen a sfficien condiion for a kind of ighness of solions of sochasic differenial eqaions (SDE) describing finie-paricle sysems, and prove ha he i poins solve he corresponding ISDE. This implies, if in addiion he i ISDE enjoy a niqeness of solions, hen he fll seqence converges. We rea non-reversible case in he firs main heorem. In he second general heorem, we resric o he case of reversible paricle sysems and simplify he sfficien condiion. We dedce he second heorem from he firs. We apply he second general heorem o Airy β ineracing Brownian moion wih β = 1, 2, 4, and he Ginibre ineracing Brownian moion. The former appears in he sof-edge i of Gassian (orhogonal/niary/symplecic) ensembles in one spaial dimension, and he laer in he blk i of Ginibre ensemble in wo spaial dimensions, corresponding o a qanm saisical sysem for which he eigen-vale specra belong o non-hermiian Gassian random marices. The passage from he finie-paricle sochasic differenial eqaion (SDE) o he i ISDE is a sensiive problem becase he logarihmic poenials are long range and nbonded a infiniy. Indeed, he i ISDEs are no easily deecable from hose of finie dimensions. Or general heorems can be applied sraighforwardly o he grand canonical Gibbs measres wih Relle-class poenials sch as Lennard-Jones 6-12 poenials and and Riesz poenials. 1. Inrodcion. Ineracing Brownian moion in infinie dimensions is prooypical of diffsion processes of infiniely many paricle sysems, iniiaed by Lang [12], [13], followed by Friz [3], Tanemra [3], and ohers. Typically, ineracing Brownian moion X = (X i ) i wih Relle-class (ranslaion invarian) ineracion Ψ and inverse emperare β is given by 21 Mahemaics Sbjec Classificaion. Primary 6B2; Secondary 6H1. Key Words and Phrases. random marix heory, infinie-dimensional sochasic differenial eqaions, ineracing Brownian moions, Airy poin processes, he Ginibre poin process, logarihmic poenial, finie-paricle approximaions. The firs ahor is sppored by Gran-in-Aid for JSPS JSPS Research Fellowships (o. 15J391). The second ahor is sppored in par by a Gran-in-Aid for Scenic Research (KIBA-A, o ; KIBA-A, o. 16H2149; KIBA-S, o. 16H6338) from he Japan Sociey for he Promoion of Science.

2 922 Y. Kawamoo and H. Osada dx i = db i β 2 Ψ(X i X j )d (i ). (1.1) j;j i Here an ineracion Ψ is called Relle-class if Ψ is sper sable in he sense of Relle, and inegrable a infiniy [28]. The sysem X is a diffsion process wih sae space S (R d ), and has no naral invarian measres. Indeed, sch a measre ˇµ, if exiss, is informally given by ˇµ = 1 Z e β (i,j); i<j Ψ(x i x j ) k dx k, (1.2) which canno be jsified as i is becase of he presence of an infinie prodc of Lebesge measres. To rigorize he expression (1.2), he Dobrshin Lanford Relle (DLR) framework inrodces he noion of a Gibbs measre. A poin process µ is called a Ψ-canonical Gibbs measre if i saisfies he DLR eqaion: for each m and µ-a.s. ξ = i δ ξ i µ m r,ξ(ds) = 1 Zr,ξ m e β{ m i<j, s i,s j Sr Ψ(s i s j )+ m s i Sr,ξ j Sr c Ψ(s i ξ j )} m k=1 ds k, (1.3) where s = i δ s i, S r = { x r}, π r (s) = s( S r ), and ξ is he oer condiion. Frhermore, µ m r,ξ denoes he reglar condiional probabiliy: µ m r,ξ(ds) = µ(π r (s) ds s(s r ) = m, π c r(s) = π c r(ξ)). Then µ is a reversible measre of he delabeled dynamics X sch ha X = i δ X i. If he nmber of paricles is finie, say, hen SDE (1.1) becomes dx, i = db i β { Φ (X, i ) + 2 j;j i Ψ(X, i } X, j ) d (1 i ), (1.4) where Φ is a confining free poenial vanishing zero as goes o infiniy. The associaed labeled measre is hen given by ˇµ = 1 Z e β{ Φ (x i)+ (i,j); i<j Ψ(xi xj)} k=1 dx k. (1.5) The relaion beween (1.4) and (1.5) is as follows. We firs consider he diffsion process associaed wih he Dirichle form wih domain D ˇµ on L 2 ((R d ), ˇµ ), called he disored Brownian moion, sch ha 1 E ˇµ (f, g) = i f i g ˇµ (dx ), (R d ) 2 where i = ( / x ij ) d j=1, x = (x 1,..., x ) (R d ), and denoes he inner prodc in R d. The generaor Lˇµ of E ˇµ is hen given by

3 Finie-paricle approximaions for ineracing Brownian paricles 923 E ˇµ (f, g) = ( Lˇµ f, g) L2 ((R d ),ˇµ ). Inegraion by pars yields he represenaion of he generaor of he diffsion process sch ha Lˇµ = 1 2 β 2 { Φ (x i ) + j; j i } Ψ(x i x j ) i, which ogeher wih Iô formla yields SDE (1.4). For a finie or infinie seqence x = (x i ), we se (x) = i δ x i and call a delabeling map. For a poin process µ, we say a measrable map l = l(s) defined for µ-a.s. s wih vale S { =1 S } is called a label wih respec o µ if l(s) = s. Le l be a label wih respec o µ. We denoe by l m and l,m he firs m-componens of hese labels, respecively. We ake Φ sch ha he associaed poin process µ = ˇµ 1 converges weakly o µ: µ = µ weakly. (1.6) The associaed delabeling X = δ X, i is reversible wih respec o µ. The labeled process X = (X i ) and X = (X, i ) can be recovered from X and X by aking siable iniial labels l and l, respecively. Choosing he labels in sch a way ha for each m µ l 1,m = µ l 1 m weakly, (1.7) we have he convergence of labeled dynamics X o X sch ha for each m (X,1,..., X,m ) = (X 1,..., X m ) in law in C([, ); (R d ) m ). (1.8) We expec his convergence becase of he absole convergence of he drif erms in (1.1) and energy in he DLR eqaion (1.3) for well-behaved iniial disribions alhogh i sill reqires some work o jsify his rigorosly even if Ψ C(R 3 d ) [12]. If we ake logarihmic fncions as ineracion poenials, hen he siaion changes drasically. Consider he sof-edge scaling i of Gassian (orhogonal/niary/ symplecic) ensembles. Then he -labeled densiy is given by ˇµ Airy,β(dx ) = 1 Z { i<j } { x i x j β exp β /6 x k }dx 2 (1.9) k=1 and he associaed -paricle dynamics described by SDE dx,i = db i + β 2 j=1, j i X,i 1 X,j d β 2 { 1/3 + 1 X,i 2 1/3 }d. (1.1) The correspondence beween (1.9) and (1.1) is ransparen and same as above. Indeed,

4 924 Y. Kawamoo and H. Osada we firs consider disored Brownian moion (Dirichle spaces wih ˇµ Airy,β as a common ime change and energy measre), hen we obain he generaor of he associaed diffsion process by inegraion by pars. SDE (1.1) hs follows from he generaor immediaely. I is known ha he hermodynamic i µ Airy,β of he associaed poin process µ Airy,β exiss for each β > [27]. Is m-poin correlaion fncion is explicily given as a deerminan of cerain kernels if β = 1, 2, 4 [1], [15]. Indeed, if β = 2, hen he m-poin correlaion fncion of he i poin process µ Airy,2 is ρ m Ai,2(x m ) = de[k Ai,2 (x i, x j )] m i,j=1, where K Ai,2 is he coninos kernel sch ha, for x y, K Ai,2 (x, y) = Ai(x)Ai (y) Ai (x)ai(y). x y We se here Ai (x) = dai(x)/dx and denoe by Ai( ) he Airy fncion given by Ai(z) = 1 dk e i(zk+k3 /3), z R. 2π R For β = 1, 4 similar expressions in erms of he qaernion deerminan are known [1], [15]. From he convergence of eqilibrim saes, we may expec he convergence of solions of SDEs (1.1). The divergence of he coefficiens in (1.1) and he very long-range nare of he logarihmic ineracion however prove o be problemaic. Even an informal represenaion of he i coefficiens is nonrivial b has been obained in [26]. Indeed, he i ISDEs are given by dx i = db i + β { 2 r X j <r,j i 1 X i X j } ϱ(x) x <r x dx d (i ). (1.11) Here ϱ(x) = 1 (,) (x) x, which is he shifed and rescaled semicircle fncion a he righ edge. As an applicaion of or main heorem (Theorem 2.2), we prove he convergence (1.8) of solions from (1.1) o (1.11) for {µ Airy,β } wih β = 2. We also prove ha he i poins of solions of (1.1) saisfy ISDE (1.11) wih β = 1, 2, 4. For general β 1, 2, 4, he exisence and niqeness of solions of (1.11) is sill an open problem. Indeed, he proof in [26] relies on a general heory developed in [18], [19], [2], [21], [25], which redces he problem o he qasi-gibbs propery and he exisence of he logarihmic derivaive of he eqilibrim sae. These key properies are proved only for β = 1, 2, 4 a presen. We refer o [2], [21] for he definiion of he qasi-gibbs propery and Definiion 2.1 for he logarihmic derivaive. Anoher ypical example is he Ginibre ineracing Brownian moion, which is an infinie-paricle sysem in R 2 (narally regarded as C), whose eqilibrim sae is he Ginibre poin process µ gin. The m-poin correlaion fncion ρ m gin wih respec o Gassian measre (1/π)e x 2 dx on C is hen given by

5 Finie-paricle approximaions for ineracing Brownian paricles 925 ρ m gin(x m ) = de[e x i x j ] m i,j=1. The Ginibre poin process µ gin is he hermodynamic i of -paricle poin process µ gin whose labeled measre is given by ˇµ gin(dx ) = 1 Z i<j The associaed -paricle SDE is hen given by dx,i = db i X,i d + j=1,j i We shall prove ha he i ISDEs are and dx i = db i + x i x j 2 e x i 2 dx. X,i X,i r X i Xj <r,j i dx i = db i X i d + r X j <r,j i X,j X,j 2 d (1 i ). (1.12) X i X j X i X j d (i ) (1.13) 2 X i X j X i X j d (i ). (1.14) 2 In [19], [25], i is proved ha hese ISDEs have he same pahwise niqe srong solion for µ gin l 1 -a.s. s, where l is a label and s is an iniial poin. As an example of applicaions of or second main heorem (Theorem 2.2), we prove he convergence of solions of (1.12) o hose of (1.13) and (1.14). This example indicaes again he sensiiviy of he represenaion of he i ISDE. Sch varieies of he i ISDEs are a resl of he long-range nare of he logarihmic poenial. The main prpose of he presen paper is o develop a general heory for finieparicle convergence applicable o logarihmic poenials, and in pariclar, he Airy and Ginibre poin processes. Or heory is also applicable o essenially all Gibbs measres wih Relle-class poenials sch as he Lennard-Jones 6-12 poenial and Riesz poenials. In he firs main heorem (Theorem 2.1), we presen a sfficien condiion for a kind of ighness of solions of sochasic differenial eqaions (SDE) describing finieparicle sysems, and prove ha he i poins solve he corresponding ISDE. This implies, if in addiion he i ISDE enjoy niqeness of solions, hen he fll seqence converges. We rea non-reversible case in he firs main heorem. In he second main heorem (Theorem 2.2), we resric o he case of reversible paricle sysems and simplify he sfficien condiion. Becase of reversibiliy, he sfficien condiion is redced o he convergence of logarihmic derivaive of µ wih marginal assmpions. We shall dedce Theorem 2.2 from Theorem 2.1 and apply Theorem 2.2 o all examples in he presen paper.

6 926 Y. Kawamoo and H. Osada If Ψ(x) = log x, β = 2 and d = 1, here exiss an algebraic mehod o consrc he associaed sochasic processes [7], [8], [9], [1], and o prove he convergence of finie-paricle sysems [24], [23]. This mehod reqires ha ineracion Ψ is he logarihmic fncion wih β = 2 and depends crcially on an explici calclaion of space-ime deerminanal kernels. I is hs no applicable o β 2 even if d = 1. As for Sine β poin processes, Tsai proved he convergence of finie-paricle sysems for all β 1 [31]. His mehod relies on a copling mehod based on monooniciy of SDEs, which is very specific o his model. The organizaion of he paper is as follows: In Secion 2, we sae he main heorems (Theorem 2.1 and Theorem 2.2). In Secion 3, we prove Theorem 2.1. In Secion 4, we prove Theorem 2.2 sing Theorem 2.1. In Secion 5, we presen examples. 2. Se p and he main heorems Configraion spaces and Campbell measres. Le S be a closed se in R d whose inerior S in is a conneced open se saisfying S in = S and he bondary S having Lebesge measre zero. A configraion s = i δ s i on S is a Radon measre on S consising of dela masses. We se S r = {s S ; s r}. Le S be he se consising of all configraions of S. By definiion, S is given by { S = s = } δ si ; s(s r ) < for each r. i By convenion, we regard he zero measre as an elemen of S. We endow S wih he vage opology, which makes S a Polish space. S is called he configraion space over S and a probabiliy measre µ on (S, B(S)) is called a poin process on S. A symmeric and locally inegrable fncion ρ n : S n [, ) is called he n-poin correlaion fncion of a poin process µ on S wih respec o he Lebesge measre if ρ n saisfies A k 1 1 Ak m m ρ n (x 1,..., x n )dx 1 dx n = m S s(a i )! (s(a i ) k i )! dµ for any seqence of disjoin bonded measrable ses A 1,..., A m B(S) and a seqence of naral nmbers k 1,..., k m saisfying k k m = n. When s(a i ) k i <, according o or inerpreaion, s(a i )!/(s(a i ) k i )! = by convenion. Hereafer, we always consider correlaion fncions wih respec o Lebesge measres. A poin process µ x is called he redced Palm measre of µ condiioned a x S if µ x is he reglar condiional probabiliy defined as µ x = µ( δ x s({x}) 1). A Radon measre µ [1] on S S is called he 1-Campbell measre of µ if µ [1] is given by µ [1] (dxds) = ρ 1 (x)µ x (ds)dx. (2.1)

7 Finie-paricle approximaions for ineracing Brownian paricles Finie-paricle approximaions (general case). Le {µ } be a seqence of poin processes on S sch ha µ ({s(s) = }) = 1. We assme: (H1) Each µ has a correlaion fncion {ρ,n } saisfying for each r ρ,n (x) = ρ n (x) niformly on Sr n for all n, (2.2) sp x Sr n sp ρ,n (x) c n 1 n c2n, (2.3) where < c 1 (r) < and < c 2 (r) < 1 are consans independen of n. I is known ha (2.2) and (2.3) imply weak convergence (1.6) [2, Lemma A.1]. As in Secion 1, le l and l be labels of µ and µ, respecively. We assme: (H2) For each m, (1.7) holds. Tha is, We shall laer ake µ l 1 µ l 1,m = µ l 1 m weakly in S m. (1.7) as an iniial disribion of a labeled finie-paricle sysem. Hence (H2) means convergence of he iniial disribion of he labeled dynamics. There exis infiniely many differen labels l, and we choose a label sch ha he iniial disribion of he labeled dynamics converges. (H2) will be sed in Theorem 2.2 and Theorem 2.1. For X = (X i ) and X = (X,i ), we se X i = j i δ X j, and X, i = j i δ X,j, where X, i denoes he zero measre for = 1. Le σ, σ : S S R d2 and b, b : S S R d be measrable fncions. We inrodce he finie-dimensional SDE of X = (X,i ) wih hese coefficiens sch ha for 1 i We assme: dx,i = σ (X,i, X, i )db i + b (X,i, X, i )d, (2.4) X = s. (2.5) (H3) SDE (2.4) and (2.5) has a niqe solion for µ l 1 -a.s. s for each : his solion does no explode. Frhermore, when S is non-void, paricles never hi he bondary. We se a = σ σ and assme: (H4) σ are bonded and coninos on S S, and converge niformly o σ on S r S for each r. Frhermore, a are niformly ellipic on S r S for each r and x a are niformly bonded on S S.

8 928 Y. Kawamoo and H. Osada From (H4) we see ha a converge niformly o a := σ σ on each compac se S r S, and ha a and a are bonded and coninos on S S. There hs exiss a posiive consan c 3 sch ha a S S, x a S S, sp a S S, sp x a S S c 3. (2.6) Here S S denoes he niform norm on S S. Frhermore, we see ha a is niformly ellipic on each S r S. From hese, we expec ha SDEs (2.4) have a sb-seqenial i. {X,i X,i } = = σ( σ (X,i, X, i )db i +, X,i X, i )db i + b (X,i b (X,i, X, i )d, X, i )d. To idenify he second erm on he righ-hand side and o jsify he convergence, we make frher assmpions. As he examples in Secion 1 sgges, he idenificaion of he i is a sensiive problem, which is a he hear of he presen paper. We se he maximal modle variable X,m of he firs m-paricles by X,m = max m sp [,T ] X,i. and by L r he maximal label wih which he paricle inersecs S r ; ha is, L r = max{i { } ; X,i r for some T }. We assme he following. (I1) For each m inf P µ l 1,m (X a) = 1 (2.7) a and here exiss a consan c 4 = c 4 (m, a) sch ha for, T sp Frhermore, for each r E µ l 1 [ X,i X,i 4 ; X,m a] c 4 2. (2.8) inf P µ l 1 (L r L) = 1. (2.9) L Le µ,[1] be he one-campbell measre of µ defined as (2.1). Se c 5 (r, ) = µ,[1] (S r S). Then by (2.3) sp c 5 (r, ) < for each r. Wiho loss of generaliy, we can assme ha c 5 > for all r,. Le µ,[1] r = µ,[1] ( {S r S}). Le µ,[1] r be he probabiliy measre defined as µ,[1] r ( ) = µ,[1] ( {S r S})/c 5. Le ϖ r,s be a map from S r S o iself sch ha ϖ r,s (x, s) = (x, x s i <s δ s i ), where s = i δ s i.

9 Finie-paricle approximaions for ineracing Brownian paricles 929 Le F r,s = σ[ϖ r,s ] be he sb-σ-field of B(S r S) generaed by ϖ r,s. Becase S r is a sbse of S, we can and do regard F r,s as a σ-field on S S, which is rivial oside S r S. We se a ail-rncaed coefficien b r,s of b and heir ail pars b,ail r,s by b r,s = E µ,[1] r [b F r,s ], b = b r,s + b,ail r,s. (2.1) We can and do ake a version of b r,s sch ha b r,s(x, y) = for x S r, (2.11) b r,s(x, y) = b r+1,s(x, y) for x S r. (2.12) We nex inrodce a c-off coefficien b r,s,p of b r,s. Le b r,s,p be a coninos and F r,s -measrable fncion on S S sch ha b r,s,p(x, y) = for x S r (2.13) b r,s,p(x, y) = b r+1,s,p(x, y) for x S r 1 (2.14) and ha, for (S S) r,p = {(x, y) S r S; x y i 1/2 p for some y i }, where y = i δ y i, b r,s,p(x, y) = for (x, y) (S S) r,p+1, (2.15) b r,s,p(x, y) = b r,s(x, y) for (x, y) (S S) r,p. (2.16) The main reqiremens for b and b r,s,p are he following: (I2) There exiss a ˆp sch ha 1 < ˆp and ha for each r sp b ˆp dµ,[1] <. (2.17) S r S Frhermore, for each r, i, here exiss a consan c 6 sch ha sp sp p [ T E µ l 1 b r,s,p(x,i ], X, i ) ˆp d c 6. (2.18) We se S m r = {s ; s(s r ) = m}. Le S S m r denoe he niform norm on S S m r and se Lˆp (µ,[1] r ) = Lˆp (S r S, µ,[1] ). For a fncion f on S S m r we denoe by f = ( x ˇf, yi ˇf), where ˇf is a fncion on Sr Sr m sch ha ˇf(x, (y i ) m ) is symmeric in (y i ) m for each x and f(x, i δ y i ) = ˇf(x, (y i ) m ). We decompose b r,s as and we assme: b r,s = b r,s,p + b r,s b r,s,p (2.19) (I3) For each m, p, r, s sch ha r < s, here exiss b r,s,p sch ha b r,s,p b r,s,p S S m s =. (2.2)

10 93 Y. Kawamoo and H. Osada Moreover, b r,s,p are differeniable in x and saisfying he bonds: Frhermore, we assme for each i, r < s p sp p Eµ l 1 where b r,s is sch ha sp b r,s,p S S m s <, (2.21) sp b r,s,p b r,s p L ˆp =. (2.22) (µ,[1] r ) [ T E µ l 1 [ T {b r,s,p b r,s}(x,i ] {b r,s,p b r,s }(X i, X i ) ˆp d ], X, i ) ˆp d =, (2.23) =, (2.24) b r,s (x, y) = b r,s(x, y) for each (x, y) p (S S) c r,p. (2.25) Remark 2.1. We see ha p (S S)c r,p = {S c r S} {(x, y); x y i for all i} by definiion and b r,s (x, y) = for x S r by (2.11). The i in (2.25) exiss becase of (2.15), (2.16), and (2.2). (I4) There exiss a b ail C(S; R d ) independen of r and s S sch ha s Frhermore, for each r, i : s [ T sp E µ l 1 (b,ail r,s sp b,ail r,s b ail L ˆp =. (2.26) (µ,[1] r ) b ail )(X,i ], X, i ) ˆp d =. (2.27) We remark ha b ail is aomaically independen of r for consisency (2.16). By assmpion, b ail = b ail (x) is a fncion of x. From (2.1) and (2.19) we have b = b r,s,p + b ail + {b r,s b r,s,p} + {b,ail r,s b ail }. (2.28) In (I3) and (I4), we have assmed ha he las wo erms {b r,s b r,s,p} and {b,ail r,s b ail } in (2.28) are asympoically negligible. Under hese assmpions, we prove in Lemma 3.1 ha here exiss b sch ha for each r We assme: (I5) For each i, r b r,s b s L ˆp =. (2.29) (µ,[1] r )

11 Finie-paricle approximaions for ineracing Brownian paricles 931 [ T ] (b s Eµ l 1 r,s b)(x, i X i ) ˆp d =. (2.3) We say a seqence {X } of C([, T ]; S )-valed random variables is igh if for any sbseqence we can choose a sbseqence denoed by he same symbol sch ha {X,m } m is convergen in law in C([, T ]; S m ) for each m. Wih hese preparaions, we sae he main heorem in his secion. Theorem 2.1. Assme (H1) (H4) and (I1) (I5). Then, {X } is igh in C([, T ]; S ) and, any i poin X = (X i ) i of {X } is a solion of he ISDE dx i = σ(x i, X i )db i + {b(x i, X i ) + b ail (X i )}d. (2.31) Remark 2.2. If diffsion processes are symmeric, we can dispense wih (2.8), (2.18), (2.23), (2.24), (2.27), and (2.3) as we see in Sbsecion 2.3. Indeed, sing he Lyons-Zheng decomposiion we can derive hese from saic condiions (H4), (2.17), (2.2), (2.22), (2.26), and (2.29). We remark ha we can apply Theorem 2.1 o nonsymmeric diffsion processes by assming hese dynamical condiions Finie-paricle approximaions (reversible case). For a sbse A, we se π A : S S by π A (s) = s( A). We say a fncion f on S is local if f is σ[π K ]-measrable for some compac se K in S. For a local fncion f on S, we say f is smooh if ˇf is smooh, where ˇf(x 1,...) is a symmeric fncion sch ha ˇf(x 1,...) = f(x) for x = i δ x i. Le D be he se of all bonded, local smooh fncions on S. We wrie f L p loc (µ[1] ) if f L p (S r S, µ [1] ) for all r. Le C (S) D = { f i(x)g i (y) ; f i C (S), g i D, } denoe he algebraic ensor prodc of C (S) and D. Definiion 2.1. A R d -valed fncion d µ L 1 loc (µ[1] ) is called he logarihmic derivaive of µ if, for all φ C (S) D, d µ (x, y)φ(x, y)µ [1] (dxdy) = x φ(x, y)µ [1] (dxdy). S S S S (1) The logarihmic derivaive d µ is deermined niqely (if ex- Remark 2.3. iss). (2) If he bondary S is nonempy and paricles hi he bondary, hen d µ wold conain a erm arising from he bondary condiion. For example, if he emann bondary condiion is imposed on he bondary, hen here wold be local imeype drifs. We shall laer assme ha paricles never hi he bondary, and he above formlaion is hs sfficien in he presen siaion. (3) A sfficien condiion for he explici expression of he logarihmic derivaive of poin processes is given in [19, Theorem 45]. Using his, one can obain he logarihmic derivaive of poin processes appearing in random marix heory sch as sine β, Airy β, (β = 1, 2, 4), Bessel 2,α (1 α), and he Ginibre poin process (see

12 932 Y. Kawamoo and H. Osada Examples in Secion 5). For canonical Gibbs measres wih Relle-class ineracion poenials, one can easily calclae he logarihmic derivaive employing DLR eqaion [25, Lemma 1.1]. We assme: (J1) Each µ has a logarihmic derivaive d, and he coefficien b is given as b = 1 2 { xa + a d }. (2.32) Frhermore, he vecor-valed fncions { x a } are coninos and converge o x a niformly on each S r S, where x a is he d-dimensional colmn vecor sch ha x a (x, y) = ( d x i a 1i(x, y),..., d ) a x di(x, y). (2.33) i Remark 2.4. From (J1) we see ha he delabeled dynamics X = δ X i of X is reversible wih respec o µ. Ths (J1) relaes he measre µ wih he labeled dynamics X. For each <, X has a reversible measre. Indeed, he symmerizaion (µ l 1 ) sym of µ l 1 is a reversible measre of X as we see for ˇµ in Inrodcion, where (µ l 1 ) sym = (1/!) σ Sym() (µ l 1 ) σ 1 and Sym() is he symmeric grop of order. When =, X does no have any reversible measre in general. For example, infinie-dimensional Brownian moion B = (B i ) i on (R d ) has no reversible measres. We also remark ha he Airy β (β = 1, 2, 4) ineracing Brownian moion defined by (1.11) has a reversible measre given by µ Airy,β l 1 wih label l(s) = (s 1, s 2,...) sch ha s i > s i+1 for all i becase l gives a bijecion from (a sbse of) S o R defined for µ Airy,β -a.s. s, and hs he relaion X = l(x ) holds for all. We prove ha convergence of he logarihmic derivaive implies weak convergence of he solions of he associaed SDEs. Each logarihmic derivaive d belongs o a differen L p -space L p (µ,[1] ), and µ,[1] are mally singlar. Hence we decompose d o define a kind of L p -convergence. Le,, w : S R d and g, g, v, v : S 2 R d be measrable fncions. We se g s (x, y) = χ s (x y)v(x, y)dy + χ s (x y i )g(x, y i ), S i g s (x, y) = χ s (x y)v (x, y)dy + χ s (x y i )g (x, y i ), S i w s (x, y) = χ s (x y)}v S{1 (x, y)dy + (1 χ s (x y i ))g (x, y i ), (2.34) i where y = i δ y i and χ s C (S) is a c-off fncion sch ha χ s 1, χ s (x) = for x s + 1, and χ s (x) = 1 for x s. We assme he following.

13 Finie-paricle approximaions for ineracing Brownian paricles 933 (J2) Each µ has a logarihmic derivaive d sch ha Frhermore, we assme ha d (x, y) = (x) + g s (x, y) + w s (x, y). (2.35) (1) are in C 1 (S). Frhermore, and converge niformly o and, respecively, on each compac se in S. (2) For each s, S χ s(x y)v (x, y)dy are in C 1 (S). Frhermore, fncions S χ s(x y)v (x, y)dy and x S χ s(x y)v (x, y)dy converge niformly o S χ s(x y)v(x, y)dy and x S χ s(x y)v(x, y)dy, respecively, on each compac se in S. (3) g are in C 1 (S 2 {x y}). Frhermore, g and x g converge niformly o g and x g, respecively, on S 2 { x y 2 p } for each p >. In addiion, for each r, p sp χ s (x y) g (x, y) ˆp ρ,1 x (y)dxdy =, (2.36) x S r, x y 2 p where ρ,1 x is a one-correlaion fncion of he redced Palm measre µ x. (4) There exiss a coninos fncion w : S R sch ha sp w s (x, y) w(x) ˆp dµ,[1] =, w Lˆp s loc (S, dx). (2.37) S r S Le p be sch ha 1 < p < ˆp. Assme (H1) and (J2). Then from [19, Theorem 45] we see ha he logarihmic derivaive d µ of µ exiss in L p loc (µ[1] ) and is given by d µ (x, y) = (x) + g(x, y) + w(x). (2.38) Here g(x, y) = s g s (x, y) and he convergence of g s akes place in L p loc (µ[1] ). We now inrodce he ISDE of X = (X i ) i : dx i = σ(x i, X i )db i { xa(x i, X i ) + a(x i, X i )d µ (X i, X i )}d (2.39) X = s. (2.4) Here x a is defined similarly as (2.33). If σ is he ni marix and (J2) is saisfied, we have dx i = db i {(Xi ) + w(x i ) + g(x i, X i )}d. (2.41) In he seqel, we give a sfficien condiion for solving ISDE (2.39) (and (2.41)). Le D be he sandard sqare field on S sch ha for any f, g D and s = i δ s i D[f, g](s) = 1 2 { i i ˇf i ǧ} (s),

14 934 Y. Kawamoo and H. Osada where is he inner prodc in R d. Since he fncion i ˇf(s) i i ǧ(s), where s = (s i ) i and s = i δ s i, is symmeric in (s i ) i, we regard i as a fncion of s. We se L 2 (µ) = L 2 (S, µ) and le E µ (f, g) = D[f, g](s)µ(ds), We assme: (J3) (E µ, D µ ) is closable on L 2 (µ). S D µ = {f D L 2 (µ) ; E µ (f, f) < }. From (J3) and he local bondedness of correlaion fncions given by (H1), we dedce ha he closre (E µ, D µ ) of (E µ, D µ ) becomes a qasi-reglar Dirichle form [16, Theorem 1]. Hence, sing a general heory of qasi-reglar Dirichle forms, we dedce he exisence of he associaed S-valed diffsion (P, X) [14]. By consrcion, (P, X) is µ-reversible. If one akes µ as Poisson poin process wih Lebesge inensiy, hen he diffsion (P, X) hs obained is he sandard S-valed Brownian moion B sch ha B = i δ B i, where {Bi } i are independen copies of he sandard Brownian moions on R d. This is he reason why we call D he sandard sqare field. Le Cap µ denoe he capaciy given by he Dirichle space (E µ, D µ, L 2 (µ)) [4]. Le and assme: (J4) Cap µ ({S s.i. } c ) =. S s.i. = {s S ; s(x) 1 for all x S, s(s) = } Le Erf() = (1/ 2π) e x 2 /2 dx be he error fncion. Le S r = { x < r} as before. We assme: (J5) There exiss a Q > sch ha for each R > { } ( ) inf sp ρ,1 r (x) dx Erf =. (2.42) r S r+r (r + R)Q We wrie s i = l (s) i and assme for each r sp Erf L i>l S ( si r c3 T ) µ (ds) =. (2.43) We remark ha (2.43) is easy o check. Indeed, we prove in Lemma 4.6 ha, if s i = l (s) i is aken sch ha hen (2.43) follows from (H1) and (2.45) below. s 1 s 2, (2.44)

15 Finie-paricle approximaions for ineracing Brownian paricles 935 q sp Erf S\S q ( ) x r ρ,1 (x)dx =. (2.45) c3 T Le l be he label as before. Le X = (X i ) i be a family of solion of (2.39) saisfying X = s for µ l 1 -a.s. s. We call X saisfies µ-absole coniniy condiion if µ µ for all, (2.46) where µ is he disribion of X and µ µ means µ is absolely coninos wih respec o µ. Here X = i δ X i, for X = (X i ) i. By definiion X = {X } is he delabeled dynamics of X and by consrcion X = µ in disribion. We say ISDE (2.39) has µ-niqeness of solions in law if X and X are solions wih he same iniial disribions saisfying he µ-absole coniniy condiion, hen hey are eqivalen in law. We assme: (J6) ISDE (2.39) has µ-niqeness of solions in law. Le X be a solion of (2.4). From (2.32) we can rewrie (2.4) as dx,i = σ (X,i, X, i )db i { xa + a d }(X,i, X, i )d. (2.47) We se X,m = (X,1, X,2,..., X,m ) 1 m and X m = (X 1, X 2,..., X m ). We say {X } is igh in C([, ); S ) if each sbseqence {X } conains a sbseqence {X } sch ha {X,m } is convergen weakly in C([, ); S m ) for each m. Theorem 2.2. Assme (H1) (H4) and (J1) (J5). Assme ha X = µ l 1 in disribion. Then {X } is igh in C([, ); S ) and each i poin X of {X } is a solion of (2.39) wih iniial disribion µ l 1. Frhermore, if we assme (J6) in addiion, hen for any m X,m = X m weakly in C([, ), S m ). (2.48) Remark 2.5. To prove (2.48) i is sfficien o prove he convergence in C([, T ]; S m ) for each T. We do his in he following secions. Remark 2.6. (1) A sfficien condiion for (J3) is obained in [2], [21]. Indeed, if µ is a (Φ, Ψ)-qasi-Gibbs measre wih pper semi-coninos poenial (Φ, Ψ), hen (J3) is saisfied. This condiion is mild and is saisfied by all examples in he presen paper. We refer o [2], [21] for he definiion of qasi-gibbs propery. (2) From he general heory of Dirichle forms, we see ha (J4) is eqivalen o he non-collision of paricles [4]. We refer o [6] for a necessary and sfficien condiion of his non-collision propery of ineracing Brownian moions in finie-dimensions, which gives a sfficien condiion of non-collision in infinie dimensions. We also refer o [17] for a sfficien condiion for non-collision propery of ineracing Brownian moions in infinie-dimensions applicable o, in pariclar, deerminanal poin processes.

16 936 Y. Kawamoo and H. Osada (3) From (2.42) of (J5), we dedce ha each agged paricle X i does no explode [4], [18]. We remark ha he delabeled dynamics X = i δ Xi are µ-reversible, and hey hs never explode. Indeed, as for configraion-valed diffsions, explosion occrs if and only if infiniely many paricles gaher in a compac domain, so he explosion of agged paricle does no imply ha of he configraion-valed process. (4) I is known ha, if we sppose (H1), (J1) (J5), hen ISDE (2.39) has a solion for µ l 1 -a.s. s saisfying he non-collision and non-explosion propery [19]. Indeed, le X = (X i ) be he S -valed coninos process consising of agged paricles X i of he delabeled diffsion process X = i δ X i given by he Dirichle form of (J3). Then from (J4) and (J5) (2.42) we see X is niqely deermined by is iniial saring poin. I was proved ha X is a solion of (2.39) in [19]. Remark 2.7. Assmpion (J6) follows from ail rivialiy of µ [25], where ail rivialiy of µ means ha he ail σ-field T = r=1 σ[π Sr c ] is µ-rivial. Indeed, from ail rivialiy of µ and marginal assmpions ((E1), (F1), and (F2) in [25]), we obain (J6). Tail rivialiy holds for all deerminanal poin processes [22] and grand canonical Gibbs measres wih sfficienly small inverse emperare β >. 3. Proof of Theorem 2.1. The prpose of his secion is o prove Theorem 2.1. We assme he same assmpions as Theorem 2.1 hrogho his secion. We begin by proving (2.29). Lemma 3.1. (2.29) holds. Proof. From (H1) and (2.2), we obain b r,s,p = b r,s,p for µ [1] r -a.s. and in Lˆp ( µ [1] r ). (3.1) We nex prove he convergence of {b r,s,p } as p. oe ha b r,s,p b r,s,q L ˆp ( µ [1] r ) b r,s,p b r,s,p L ˆp ( µ [1] r ) + b r,s,p b r,s,q L ˆp ( µ [1] r ) + b r,s,q b r,s,q L ˆp ( µ [1] r ). (3.2) From (2.22) for each ϵ here exiss a p sch ha for all p, q p By (3.1) here exiss an = p,q sch ha sp b r,s,p b r,s,q L < ϵ. (3.3) ˆp (µ,[1] r ) b r,s,p b r,s,p L ˆp ( µ [1] r ) < ϵ, b r,s,q b r,s,q L ˆp < ϵ. (3.4) ( µ [1] r ) Ping (3.3) and (3.4) ino (3.2), we dedce ha {b r,s,p } p is a Cachy seqence in Lˆp ( µ [1] r ). Hence from (2.16), (2.22), and (2.25) we see p b r,s,p = b r,s in Lˆp ( µ [1] r ). (3.5)

17 Finie-paricle approximaions for ineracing Brownian paricles 937 Recall ha b r,s = E µ,[1] r [b F r,s ] by (2.1). Then, becase F r,s F r,s+1, we have From b r,s = E µ,[1] r [b F r,s ] we have From his and (2.17) we obain sp r<s sp b r,s = E µ,[1] r [b r,s+1 F r,s ]. (3.6) b r,s L ˆp ( µ,[1] r ) b L ˆp ( µ,[1] r ). b r,s L ˆp ( µ,[1] r ) Combining (2.25), (3.6) and (3.7), we have b r,s = b r,s = E µ,[1] r From (H1), (2.25), (3.7), and Fao s lemma, we see ha sp r<s b r,s L ˆp ( µ [1] r ) sp r<s sp b L <. (3.7) ˆp ( µ,[1] r ) [b r,s+1 F r,s ] = E µ[1] r [br,s+1 F r,s ]. (3.8) inf b r,s L ˆp <. (3.9) ( µ,[1] r ) From (3.8) we dedce ha {b r,s } s=r+1 is maringale in s. Applying he maringale convergence heorem o {b r,s } s=r+1 and sing (3.9), we dedce ha here exiss a b r sch ha and ha b r,s = E µ[1] r [br F r,s ] (3.1) b r,s = b r for µ [1] r -a.s. and in Lˆp ( µ [1] r ). s By he consisency of { µ [1] r } r in r, he fncion b r in (3.1) can be aken o be independen of r. This ogeher wih (3.5) complees he proof of (2.29). We proceed wih he proof of he laer half of Theorem 2.1. Recall SDE (2.4). Then X,i X,i = σ (X,i, X, i )db i + Using he decomposiion in (2.28), we see from (3.11) ha X,i X,i = + σ (X,i, X, i )db i + [ {b r,s b r,s,p} + {b,ail r,s b ail } b (X,i, X, i )d. (3.11) {b r,s,p + b ail }(X,i ] (X,i, X, i )d, X, i )d. (3.12)

18 938 Y. Kawamoo and H. Osada Le i,j = / x i,j, x i = (x i,j ) d j=1 Rd, and x m = (x i ) m (Rd ) m. Se i = ( i,j ) d j=1. Le ψ C (S m ) and a i i i ψ(x m ) = d k,l=1 a kl (x i) i,k i,l ψ(x m ). Applying he Iô formla o ψ and (3.12), and ping X,m = (X,1,..., X,m ), we dedce ha ψ(x,m ) ψ(x,m ) = + ( i ψ(x,m ) σ (X,i 1 2 a i i i ψ(x,m ) + {b r,s,p + b ail }(X,i + i ψ(x,m ) {b r,s b r,s,p}(x,i + i ψ(x,m ) {b,ail r,s, X, i, X, i )d, X, i )db i ) ) i ψ(x,m )d b ail }(X,i, X, i )d. (3.13) We se Lemma 3.2. Q r,s,p = R r,s = T T {b r,s b r,s,p}(x,i {b,ail r,s For each m, r < s, X, i d, ) b ail }(X,i, X, i ) d. p s sp sp [ E µ l 1 (Q )ˆp] r,s,p =, [ E µ l 1 (R )ˆp] r,s =. Proof. Lemma 3.2 follows from (2.23) and (2.27) immediaely. Le Ξ = S m (R d2 ) m (R d ) m and ψ C (S m ). Le F : C([, T ]; Ξ) C([, T ]; R) sch ha F (ξ, η, ζ)() = ψ(ξ()) ψ(ξ()) ζ i () i ψ(ξ())d ( 1 2 η i() i ψ(ξ()) + b ail (ξ i ()) i ψ(ξ())) d, (3.14) where ξ = (ξ i ) m, η = (η i) m, η i = (η i,kl ) d k,l=1, ζ = (ζ i) m, and i = d j=1 2 i,j. As ψ C (S m ) and b ail C(S m ) by definiion, we see ha F saisfies he following. (1) F is coninos. (2) F (ξ, η, ζ) is bonded in (ξ, η) for each ζ, and linear in ζ for each (ξ, η).

19 Finie-paricle approximaions for ineracing Brownian paricles 939 Le A,m = (A,i ) m and B,m r,s,p = (B,i r,s,p) m sch ha A,i () = a (X,i, X, i ), B,i Then we see from (3.13) (3.15) ha for each m F (X,m, A,m, B,m r,s,p) r,s,p() = b r,s,p(x,i i ψ(x,m ) σ (X,i, X, i ). (3.15), X, i )db i c 7 {Q r,s,p + R r,s}, (3.16) where c 7 = c 7 (ψ) is he consan sch ha c 7 = max m iψ S m ( A is he niform norm over A as before). We ake he i of each erm in (3.16) in he seqel. Lemma 3.3. {X,i }, {A,i } and {B,i r,s,p} are igh for each i, r, s, p. Proof. The ighness of {X,i } is clear from (I1). We noe ha { x a } is niformly bonded on S r S for each r by (H4). Hence from his and (I1) here exiss a consan c 8 independen of sch ha for all, v T E µ l 1 [ A,i () A,i (v) 4 ; sp X,i a] c 8 v 2. [,T ] By (I1) we see ha {A,i ()} is igh. Combining hese dedces he ighness of {A,i }. Recall ha B,i r,s,p() = b r,s,p(x,i, X, i ) and ha b r,s,p is F r,s -measrable by assmpion. By consrcion P µ l 1 (X,j S r for all 1 j m, T L r+s m) = 1. (3.17) Le c 9 = sp b r,s,p S S m 1. From (3.15), (2.21), (3.17), and (2.8) we see s E µ l 1 [ B,i r,s,p() B,i r,s,p(v) 4 ; sp X,i a, L r+s m] [,T ] = E µ l 1 E µ l 1 [ b r,s,p (X,i [ m j=1 c 4 9 X,j, X, i ) b r,s,p(x,i v c 4 9c 6 v 2 for all, v T. Xv,j 4 ; sp X,i a, L r+s m] [,T ], X, i v ) 4 ; sp X,i a, L r+s m] [,T ] From his, (2.7), and (2.9), we dedce he ighness of {B,i r,s,p}. Lemma 3.4. m, r, s, p. {((X,i, A,i, B,i r,s,p)) m } is igh in C([, T ], Ξ m ) for each

20 94 Y. Kawamoo and H. Osada Proof. Lemma 3.4 is obvios from Lemma 3.3. Indeed, he ighness of he probabiliy measres on a conable prodc space follows from ha of he disribion of each componen. Assmpion (I1) and Lemma 3.4 combined wih he diagonal argmen imply ha for any sbseqence of {((X,i, A,i, B,i r,s,p)) m }, p, r<s<, here exiss a convergenin-law sbseqence, denoed by he same symbol. Tha is, for each p, s, r, m, (X,i, A,i, B,i r,s,p) m = (X i, A i, B i r,s,p) m in law. (3.18) We hs assme (3.18) in he res of his secion. Le A m = (A i ) m, B,m r,s,p = (B,i r,s,p) m, and Xm = (X i ) m for X = (Xi ) i in Theorem 2.1. Lemma 3.5. For each m F (X,m, A,m, B,m r,s,p) = F (X m, A m, B m r,s,p) in law. (3.19) Moreover, A i and B i r,s,p are given by A i () = a(x i, X i ), B i r,s,p() = b r,s,p (X i, X i ). (3.2) Proof. Recall ha F (ξ, η, ζ) is coninos. Hence (3.19) follows from (3.18). By (H4) we see {a } converges o a niformly on each S r S. Then, from his, (2.2), and (3.15) we obain (3.2). Lemma 3.6. For each m = i ψ(x,m ) σ (X,i i ψ(x m ) σ(x i, X i )d ˆB i, X, i )db i in law, where ( ˆB i ) m is he firs m-componens of a (Rd ) -valed Brownian moion ( ˆB i ) i. Proof. i,k ψ(x,m ) = δ ij By he calclaion of qadraic variaion, we see d n=1 a kl(x,i σ kn(x,i, X, i, X, i )db i,n, ) i,k ψ(x,m j,l ψ(x,m ) ) i,l ψ(x,m )d. d n=1 σ ln(x,j, X, j )db j,n From (H4), we see ha a converges o a niformly on S r for each r. Hence we dedce from (I1) and ψ C (S m ) he convergence in law sch ha

21 Finie-paricle approximaions for ineracing Brownian paricles 941 = a kl(x,i, X, i ) i,k ψ(x,m ) i,l ψ(x,m )d a kl (X i, X i ) i,k ψ(x m ) i,l ψ(x m )d. Then he righ-hand side gives he qadraic variaion of σ(x, i X i )d ˆB. i This complees he proof. m iψ(x m ) We are now ready for he proof of Theorem 2.1. Proof of Theorem 2.1. [ spe µ l 1 From Lemma 3.2 and (3.16) we dedce ha F (X,m, A,m, B,m r,s,p)() sp T sp E µ l 1 i ψ(x,m ) σ (X,i, X, i )db i ˆp] [ (Q r,s,p )ˆp + (R r,s)ˆp] =: c 1 (s, p), where c 1 (s, p) = c 1 (s, p, ψ) is a consan depending on s, p, ψ. Lemma 3.5 and Lemma 3.6 o (3.16), we hen dedce ha E µ l 1 [ sp T F (Xm, A m, B m r,s,p)() i ψ(x m ) σ(x, i X i )d ˆB i ˆp] Applying c 1 (s, p). From his and (3.14), we obain ha E µ l 1 [ sp T c 1 (s, p). ψ(xm ) ψ(x m ) i ψ(x m ) σ(x i, X i )d ˆB i 1 2 a(xi, X i ) i i ψ(x m ) + b ail (X i ) i ψ(x m )d b r,s,p (X, i X i ) i ψ(x m )d ˆp] (3.21) Take ψ = ψ R C (S m ) sch ha ψ(x 1,..., x m ) = x i for { x j R; j = 1,..., m} while keeping i ψ bonded in sch a way ha Then we dedce from (3.21) ha c 1 (p, s) = sp c 1 (p, s, R) = o(p, s). R

22 942 Y. Kawamoo and H. Osada E µ l 1 [ τr τr sp Xi τ R X i σ(x, i X i )d ˆB i ˆp] {b r,s,p (X, i X i ) + b ail (X)}d i c 1 (s, p), (3.22) T where τ R is a sopping ime sch ha, for X m = (X i, X i ) m C([, T ]; (S S)m ), τ R = inf{ > ; X i R for some i = 1,..., m}. As R > is arbirary, (3.22) holds for all R >. Taking R, we hs obain [ E µ l 1 sp Xi X i σ(x, i X i )d ˆB i T ] {b r,s,p (X, i X i ) + b ail (X)}d i [ τr inf sp R Eµ l 1 Xi τ R X i σ(x, i X i )d ˆB i T τr ] {b r,s,p (X, i X i ) + b ail (X)}d i c 1 (s, p) 1/ˆp. (3.23) We noe here ha he inegrands in he firs and second lines of (3.23) are niformly inegrable becase of (3.22). Taking p, hen s in (3.23), and sing assmpions (2.24) and (2.3) we hs obain [ ] E µ l 1 sp Xi X i σ(x, i X i )d ˆB i {b(x, i X i ) + b ail (X)}d i =. T This implies for all T X i X i σ(x i, X i )d ˆB i {b(x i, X i ) + b ail (X i )}d =. (3.24) We dedce (2.31) from (3.24), which complees he proof of Theorem Proof of Theorem 2.2. Is his secion we prove Theorem 2.2 sing Theorem 2.1. (H1) (H4) are commonly assmed in Theorem 2.2 and Theorem 2.1. Hence or ask is o derive condiion (I1) (I5) from condiions saed in Theorem 2.2. From (J2) we easily dedce ha = in Lˆp loc (S, dx), (4.1) g s = g s in Lˆp loc (µ[1] ) for all s. (4.2) Lemma 4.1. µ has a logarihmic derivaive d µ in L p loc (µ[1] ), where 1 p < ˆp.

23 Finie-paricle approximaions for ineracing Brownian paricles 943 Proof. We se a general heory developed in [19]. (H1) corresponds o (4.1) and (4.2) in [19]. (4.1), (4.2), (2.35), and (2.37) correspond o (4.15), (4.3), (4.29), and (4.31) in [19]. Then all he assmpions of [19, Theorem 45] are saisfied. We hs dedce Lemma 4.1 from [19, Theorem 45]. Le {X } be a seqence of solions in (2.4) and (2.5). We se he m-labeling X,[m] = ( X,1,..., X,m, j=1+m δ X,j ). (4.3) I is known [18], [19] ha X,[m] is a diffsion process associaed wih he Dirichle form E µ,[m] on L 2 (S m S, µ,[m] ) sch ha E µ,[m] (f, g) = S m S { 1 m } i f i g + D[f, g]dµ,[m], (4.4) 2 where he domain D [m] is aken as he closre of D [m] = C (S m ) D. oe ha he coordinae fncion x i = x i 1 is locally in D [m]. From his we can regard {X,i } as a Dirichle process of he m-labeled diffsion X associaed wih he Dirichle space as above. In oher words, we can wrie X,i X,i = f i (X ) f i (X ) =: A [f i], where f i (x, s) = x i 1, x i R d, and x = (x j ) m j=1 (Rd ) m. By he Fkshima decomposiion of X,i, here exis a niqe coninos local maringale addiive fncional M,i = {M,i } and an addiive fncional of zero energy,i = {,i X,i X,i = M,i +,i. } sch ha We refer o [4, Chaper 5] for he Fkshima decomposiion. Becase of (2.4), we hen have M,i = σ (X,i, X, i )db, i,i = b (X,i, X, i )d. Lemma 4.2. Le r T : C([, T ]; S) C([, T ]; S) be sch ha r T (X) = X T. Sppose ha X,[m] = µ,[m] in law. Then X,i X,i = 1 2 M,i (M,i T (r T ) M,i T (r T )) almos srely. (4.5) Proof. Applying he Lyons-Zheng decomposiion [4, Theorem 5.7.1] o addiive fncionals A [fi] for 1 i m, we obain (4.5). Lemma 4.3. (I1) holds.

24 944 Y. Kawamoo and H. Osada Proof. Alhogh M,i is a d-dimensional maringale by definiion, we assme d = 1 here and prove only his case for simpliciy. The general case d 1 can be proved in a similar fashion. Le c 3 be he consan in (2.6) (nder he assmpion d = 1). Then we noe ha for v M,i M,i v = v A,i ()d c 3 ( v) (4.6) We begin by proving (2.8). From a sandard calclaion of maringales and (4.6), we obain E µ l 1 [ M,i M,i v 4 ] = E µ l 1 = 3E µ l 1 c 11 v 2, [ B M,i B M,i v 4 ] [ M,i M,i v 2] where c 11 = 3c 2 3 and {B } is a one-dimensional Brownian moion. Applying he same calclaion o M,i T (r T ) M,i T (r T ), we have E µ l 1 [ M,i T (r T ) M,i T (r T ) 4 ] c 11 2 for each, T. (4.7) Combining (4.5) and (4.7) wih he Lyons-Zheng decomposiion (4.5), we hs obain E µ l 1 [ X,i X,i 4 ] 2c 11 2 for each, T. (4.8) Taking a sm over i = 1,..., m in (4.8), we dedce (2.8). We nex prove (2.7). From (4.5) we have 2 X,i X,i M,i + M,i T (r T ) M,i T (r T ) almos srely. From his and a represenaion heorem of maringales, we obain P µ l 1 ( sp X,i X,i a) [,T ] P µ l 1 ( sp [,T ] M,i = 2P µ l 1 ( sp M,i a) [,T ] a) + P µ l 1 ( sp M,i T (r T ) M,i T (r T ) a) [,T ] = 2P µ l 1 ( sp B M,i a). (4.9) [,T ] A direc calclaion shows ( ) P µ l 1 ( sp B M,i a) P µ l 1 a ( sp B [,T ] [, a) Erf c 3 T ] c3 T (4.1) From (4.9), (4.1), and (H2), we obain (2.7). We proceed wih he proof of (2.9). Similarly as (4.9) and (4.1), we dedce

25 P µ l 1 ( inf Finie-paricle approximaions for ineracing Brownian paricles 945 [,T ] X,i r) P µ l 1 ( sp X,i X,i X,i r) [,T ] 2P µ l 1 ( sp 2 Erf S [,T ] ( si r c3 T M,i X,i r) ) µ (ds), (4.11) where s i = l (s) i. We noe ha X,i = s i by consrcion. From (4.11) and (2.43), we dedce sp P µ l 1 (L r > L) sp 2 sp i>l i>l (L ). P µ l 1 ( inf [,T ] X,i r) S Erf ( si r c3 T ) µ (ds) This complees he proof. Lemma 4.4. (I2) holds. Proof. ha [ T E µ l 1 (2.17) follows from (4.1), (4.2), and (2.37). For each i we dedce b r,s,p(x,i, X, i ] ) ˆp d = E µ l 1 [ T E µ l 1 [ = E µ,[1] [ T T b r,s,p(x,i b r,s,p(x,i ], X, i ) ˆp d ], X, i ) ˆp d ] b r,s,p(x,[1] ) ˆp d. (4.12) Diffsion process X,[1] in (4.3) wih m = 1 given by he Dirichle form E µ,[1] in (4.4) is µ,[1] -symmeric. Hence we see ha for all T E µ,[1] [ b r,s,p(x,[1] ) ˆp ] b r,s,p ˆp dµ,[1]. S S This yields T d E µ,[1] [ b r,s,p(x,[1] ) ˆp ] T b r,s,p ˆp dµ,[1]. (4.13) S S From (4.12) and (4.13) we obain (2.18). Lemma 4.5. (I3) (I5) hold. Proof. Condiions (2.2), (2.21), and (2.22) follow from (J1), (J2), (I1), (I2),

26 946 Y. Kawamoo and H. Osada and (2.34). Similarly, as Lemma 4.4, we obain for each i [ T E µ l 1 (b r,s b r,s,p)(x,i ], X, i ) ˆp d T b r,s b r,s,p ˆp dµ,[1]. (4.14) S S Hence (2.23) follows from (4.14) and (2.22). (2.24) follows from (3.5) and an ineqaliy similar o (4.14). We have hs obained (I3). Condiion (2.26) follows from (J1) and (J2). Similarly, as Lemma 4.4, we obain for each i [ T E µ l 1 (b,ail r,s b ail )(X,i ], X, i ) ˆp d T b,ail r,s b ail ˆp dµ,[1]. S S This ogeher wih (2.26) implies (2.27). Hence we have (I4). Similarly as Lemma 4.4, we obain (2.3) from (2.29). We have hs obained (I5). Proof of Theorem 2.2. (I1) (I5) follows from Lemma 4.3 Lemma 4.5. Hence we dedce Theorem 2.2 from Theorem 2.1. We finally presen a sfficien condiion of (2.43). Lemma 4.6. Assme (H1) and (2.45) for each r as Secion 2. We ake he label l as (2.44). Then (2.43) holds. Proof. Le c 12 = c 12 () be sch ha c 12 = Erf S ( x r c3 T ) ρ,1 (x)dx. Le c 13 = sp c 12 (). Then from (H1) and (2.45), we see ha for each large r ( ) ( ) x r x r c 13 Erf ρ,1 (x)dx + sp Erf ρ,1 (x)dx S r c3 T S\S r c3 T <. (4.15) From (H1) we see ha {µ } converges o µ weakly. Hence {µ } is igh. This implies ha here exiss a seqence of increasing seqences of naral nmbers a n = {a n (m)} m=1 sch ha a n < a n+1 and ha for each m n sp µ (s(s m ) a n (m)) =. Wiho loss of generaliy, we can ake a n (m) > m for all m, n. Then from his, we see ha here exiss a seqence {p(l)} L converging o sch ha p(l) < L for all L and ha L sp µ (s(s p(l) ) L) =. (4.16) Recall ha he label l (s) = (s i ) i saisfies s 1 s 2. Using his, we divide he se S as in sch a way ha

27 Finie-paricle approximaions for ineracing Brownian paricles 947 {s L S p(l) } and {s L S p(l) }. Then s {s L S p(l) } if and only if s(s p(l) ) L. Hence we easily see ha i>l S Erf( s i r c3 T )µ (ds) c 12 ()µ ({s(s p(l) ) L}) + Taking he is on boh sides, we obain ( si r sp Erf L c3 T c 13 L i>l S ) µ (ds) sp µ ({s(s p(l) ) L}) + L S\S p(l) Erf sp Erf S\S p(l) ( ) x r ρ,1 (x)dx. c3 T ( ) x r ρ,1 (x)dx. c3 T Applying (4.15) and (4.16) o he second erm, and (2.45) o he hird, we dedce (2.43). 5. Examples. The finie-paricle approximaion in Theorem 2.2 conains many examples sch as Airy β poin processes (β = 1, 2, 4), Bessel 2,α poin process, he Ginibre poin process, he Lennard-Jones 6-12 poenial, and Riesz poenials. The firs hree examples are relaed o random marix heory and he ineracion Ψ(x) = log x, he logarihmic fncion. We presen hese in his secion. For his we shall confirm he assmpions in Theorem 2.2, ha is, assmpions (H1) (H4) and (J1) (J6). Assmpion (H1) is saisfied for he firs hree examples [15], [29]. As for he las wo examples, we assme (H1). We also assme (H2). (H3) can be proved in he same way as given in [25]. In all examples, a is always a ni marix. Hence i holds ha (H4) is saisfied and ha (2.32) in (J1) becomes b = d /2. From his we see ha SDEs (2.47) and (2.39) become dx,i = db,i d (X,i, X, i ) d (1 i ), (5.1) dx i = db i dµ (X i, X i ) d (i ), (5.2) where d µ is he logarihmic derivaive of µ given by (2.38). Assmpion (J6) for he firs hree examples wih β = 2 can be proved in he same way as [25] as we explained in Remark 2.7. Ths, in he res of his secion, or ask is o check assmpions (J2) (J5) The Airy β ineracing Brownian moion (β = 1, 2, 4). Le µ Airy,β and µ Airy,β be as in Secion 1. Recall SDEs (1.1) and (1.11) in Secion 1. Le X = (X,i ) and X = (Xi ) i be solions of

28 948 Y. Kawamoo and H. Osada dx,i = db i + β 2 dx i = db i + β 2 r 1 X,j d β { 1/ { 1 ϱ(x) x <r x dx X,i j=1, j i X j <r,j i X i X j X,i 1/3 } d, (1.1) } d (i ). (1.11) Proposiion 5.1. If β = 1, 4, hen each sb-seqenial i of solions X of (1.1) saisfies (1.11). If β = 2, hen he fll seqence converges o (1.11). Proof. Condiions (J2) (J5) oher han (2.36) can be proved in he same way as given in [26]. In [26], we ake χ s (x) = 1 Ss (x); is adapaion o he presen case is easy. We consider esimaes of correlaion fncions sch ha inf ρ,1 Airy,β (x) c 14 for all x S r, (5.3) sp ρ,2 Airy,β (x, y) c 15 x y for all x, y S r, (5.4) where c 14 (r) and c 15 (r) are posiive consans. The firs esimae is rivial becase ρ,1 Airy,β converges o ρ 1 Airy,β niformly on S r and, all hese correlaion fncions are coninos and posiive. The second esimae follows from he deerminanal expression of he correlaion fncions and bonds on derivaive of deerminanal kernels. Esimaes needed for he proof can be fond in [26] and he deail of he proof of (5.4) is lef o he reader. Eqaion (2.36) follows from (5.3) and (5.4). Indeed, he inegral in (2.36) is aken on he bonded domain and he singlariy of inegral of g (x, y) = β/(x y) near {x = y} is logarihmic. Frhermore, he one-poin correlaion fncion ρ,1 Airy,β,x of he redced Palm measre condiioned a x is conrolled by he pper bond of he wo-poin correlaion fncion and he lower bond of one-poin correlaion fncion becase ρ,1 Airy,β,x ρ,2 Airy,β (x, y) (y) = ρ,1 Airy,β (x). Using hese facs, we see ha (5.3) and (5.4) imply (2.36) The Bessel 2,α ineracing Brownian moion. Le S = [, ) and α [1, ). We consider he Bessel 2,α poin process µ bes,2,α and heir -paricle version. The Bessel 2,α poin process µ bes,2,α is a deerminanal poin process wih kernel K bes,2,α (x, y) = J α( x) yj α( y) xj α( x)j α ( y) 2(x y) xjα+1 ( x)j α ( y) J α ( x) yj α+1 ( y) =, (5.5) 2(x y) where J α is he Bessel fncion of order α [29], [5]. The densiy m α (x)dx of he associaed -paricle sysems µ bes,2,α is given by

29 Finie-paricle approximaions for ineracing Brownian paricles 949 m α (x) = 1 Zα e x i/4 j=1 x α j x k x l 2. (5.6) k<l I is known ha µ bes,2,α is also deerminanal [29, p.945] and [2, p.91] The Bessel 2,α ineracing Brownian moion is given by he following [5]. { dx,i = db i { α dx i = db i + 2X i + j i α 2X,i 1 + X i X j j=1,j i X,i 1 X,j This appears a he hard edge of one-dimensional sysems. } d (1 i ), (5.7) } d (i ). (5.8) Proposiion 5.2. Assme α > 1. Then (2.48) holds for (5.7) and (5.8). Proof. (J2) (J5) excep (2.43) are proved in [5]. We easily see ha he assmpions of Lemma 4.6 hold and yield (2.43). We hs obain (J5). Remark 5.1. There exis oher naral ISDEs and -paricle sysems relaed o he Bessel poin processes. They are he non-colliding sqare Bessel processes and heir sqare roo. The non-colliding sqare Bessel processes are reversible o he Bessel 2,α poin processes, b he associaed Dirichle forms are differen from he Bessel 2,α ineracing Brownian moion. Indeed, he coefficiens a and a in Secion 2 are aken o be a (x.y) = a(x.y) = 4x. On he oher hand, each sqare roo of he non-colliding Bessel processes is no reversible o he Bessel 2,α poin processes, b has he same ype of Dirichle forms as he Bessel 2,α ineracing Brownian moion. In pariclar, he coefficiens a and a in Secion 2 are aken o be a (x.y) = a(x.y) = 1. Tha is, hey are consan ime change of disored Brownian moion wih he sandard sqare field. We refer o [1], [11], [24] for hese processes. For reader s convenience we provide an ISDE describing he non-colliding sqare Bessel processes and heir sqare roo. We noe ha SDE (5.1) is a consan ime change of ha in [11], [24]. Le Y = (Y,i ) and Y = (Y i ) i be he non-colliding sqare Bessel processes. Then for 1 i { dy,i = 2 Y,i db i + 4 Y,i 8 + α { α + 1 dy i = 2 Y i db i Y i Y i Y j j i j=1,j i Y,i Y,i Y,j } d, (5.9) } d (i ). (5.1) Le Z = (Z,i ) and Z = (Zi ) i be sqare roo of he non-colliding sqare Bessel processes. Then applying Iô formla we obain from (5.9) and (5.1)