Uniqueness of Fokker-Planck equations for spin lattice systems (I): compact case

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1 Semigroup Forum (213) 86: DOI 1.17/s y RESEARCH ARTICLE Uniqueness of Fokker-Planck equations for spin lattice systems (I): compact case Ludovic Dan Lemle Ran Wang Liming Wu Received: 8 August 212 / Accepted: 15 October 212 / Published online: 9 November 212 Springer Science+Business Media New York 212 Abstract The main purpose of this paper is to prove the uniqueness of the Fokker- Planck equation for continuous lattice systems when the spin space is a compact, connected, C -Riemannian manifold M. Keywords Lattice systems Fokker-Planck equations Diffusion equations 1 Introduction and main results In recent years there has been a growing interest in the investigation of lattice systems when the spin space of every particle (single spin space) is a Riemannian manifold (e.g. [4, 9, 18]). Equilibrium states of such systems are described by Gibbs measures on infinite product of manifolds and the stochastic dynamics corresponding to these states is given by a semigroup. An important problem is the uniqueness of this Communicated by Abdelaziz Rhandi. L.D. Lemle ( ) Engineering Faculty of Hunedoara, Politehnica University of Timişoara, Hunedoara, Romania dan.lemle@fih.upt.ro R. Wang School of Mathematical sciences, University of Science and Technology of China, Hefei, Anhui Province 2326, China wangran@ustc.edu.cn L. Wu Institute of Applied Math., Chinese Academy of Sciences, 119, Beijing, China Li-Ming.Wu@math.univ-bpclermont.fr L. Wu Laboratoire de Math. CNRS-UMR 662, Université Blaise Pascal, Aubière, France

2 584 L.D. Lemle dynamics and we refer the reader to [3, 11 17] for the uniqueness in the sense of martingale problem and [1, 2] for the essential self-adjointness. In this paper we further develop a method to prove the uniqueness of solutions of Fokker-Planck equation for continuous spin lattice systems. Fokker-Planck equations in infinite dimensions has been studied intensively in recent years (see e.g. [5 7] and references therein). Our framework is as follows: The Lattice. The lattice used in our models will be the d-dimensional lattice Z d for some d N.Fork Z d, we shall use the Euclidean norm k = k R d generated by the natural embedding Z d R d.givenλ Z d,letλ c = Z d Λ be the complement of Λ, Λ the cardinality of Λ and Λ + k the translation {j + k ; j Λ} of Λ by k Z d. Finally, we will occasionally use the notation Λ Z d to mean that Λ Z d and Λ <. The Spin Space. The single spin space for our model will be a compact, connected, C -Riemannian manifold M equipped with Riemannian metric ρ. Also,we use T(M) to denote the tangent bundle over M. The Riemannian structure on M defines in the usual way the gradient :C 1 (M) T(M)(see [15]). The Configuration Space. Our configuration space will be the product manifold E = M Zd η = (η i ) i Z d endowed with the product topology. Given Λ Z d,let E η η Λ = (η i ) i Λ M Λ be the natural projection taking E onto M Λ. When Λ ={i}, we shall denote η {i} by η i for simplicity. For each i Z d,thelattice shift transformation θ i : E E is ( θ i η ) j = η i+j for any η E, j Z d. Given a family of potentials V ={V Λ : Λ Z d }, which satisfy the following assumptions: (H 1 ) Differentiability: (H 2 ) Shift-invariance: V Λ C 2( M Λ) for any V Λ V. V Λ θ i = V Λ+i for any i Z d,v Λ V. Here and after V Λ will be regarded as a function on M Zd such that V Λ (η) = V Λ (η Λ ). (H 3 ) Finite range of interactions: there exists some constant R > such that V Λ for any Λ Z d with diam(λ) > R. For each Λ Z d,letd Λ = Cb (MΛ ), and let D = Cb (E) = Λ Z d D Λ (1.1)

3 Uniqueness of Fokker-Planck equations for spin lattice systems (I) 585 be the test functions space. Consider the operator LF = ( Δi F i H, i F ), F D (1.2) i Z d where i and Δ i denote the gradient and Laplace-Beltrami operator (see [15]) with respect to the variable η i M respectively, and the Hamiltonian H : E R is formally given by H(η)= V Λ (η Λ ) and i H(η)= i V Λ (η Λ ). Λ Z d Λ Z d The main result of this paper is Theorem 1.1 Assume (H 1 ) (H 3 ). Then for any initial bounded measure ν on E, the Fokker-Planck equation { t ν t = L ν t (1.3) ν = ν has a unique weakly continuous solution ν t, satisfying ν t TV ce at for all t and for some constants c 1, a>. More precisely, there is a unique measurevalued function t ν t which satisfies the following properties: (i) ν t is weakly continuous, i.e., for any F D, t ν t (F ) = ν t,f = Fdν t is continuous; (ii) For any F D, it holds that ν t ν,f = t E ν s, LF ds, t ; (1.4) (iii) There are some constants c 1,a > satisfying that ν t TV ce at for all t. Notice that if {X(t)} t is a solution to the martingale problem (L, D,ν), i.e., for every F D, M t (F ) := F ( X(t) ) F ( X() ) t LF ( X(s) ) ds is a ν-local martingale (see [14]), ν t = P(X(t) ) satisfies the Fokker-Planck equation ν t,f ν,f = t ν s, LF ds, F D.

4 586 L.D. Lemle Combined with the famous martingale uniqueness criterion of Stroock-Varadhan [1, Chapter 4, Theorem 4.2] and the equivalent conditions in Wu and Zhang [19, Theorem 2.1], we have the following two corollaries: Corollary 1.2 Assume (H 1 ) (H 3 ). Then for any initial probability measure ν on E, there exists a unique solution to the martingale problem (L, D,ν). When M is the circle, the martingale uniqueness was proved by Holley and Stroock [12]. For general compact Riemannian manifold M, the result was proved in the thesis of L. Clemens [8], see [13, Theorem 2.1]. (Thanks F.Y. Wang for communications.) Corollary 1.3 Assume (H 1 ) (H 3 ). Then there exists a unique C -semigroup {T t } t on C b (E) such that its generator extends L. This paper is organized as follows. In the next section we prove the existence of solution to the martingale problem (L, D,ν) by means of finite volume approximation, using a crucial compactness result in the general theory developed in Ethier and Kurtz [1]. The last section is devoted to the proof of Theorem The existence of solution to the martingale problem (L, D,ν) We shall prove the existence of solution to the martingale problem (L, D,ν),by means of finite volume approximation and using the next crucial compactness result in the general theory developed by Ethier and Kurtz: Lemma 2.1 [1, Chap. 3, Theorem 9.1] A sequence of E-valued càdlàg stochastic processes {X n ( )} n N is relatively compact in law in D(R +,E) (the space of E- valued càdlàg functions), if the following conditions are satisfied: (i) (Compact containment condition) For any ε> and T>, there exists a compact set K E such that inf n 1 P( X n (t) K, t [,T] ) 1 ε. (ii) For any F D, {F(X n ( ))} n N is relative compact in law in D(R +, R). Now we prove the existence of solution to the martingale problem. Theorem 2.2 For any initial probability measure ν on E, there exists a continuous process {X(t)} t valued in E, which is a solution to the martingale problem (L, D,ν). Proof We shall prove it in several steps by means of finite volume approximation. Step 1. Finite volume continuous spin system. For each n N,let Λ n := [ n, n] d

5 Uniqueness of Fokker-Planck equations for spin lattice systems (I) 587 and where L n F := Δ i F i H Λn, i F, i Λ n i Λ n F D Λn H Λn (η) = V Λ (η Λ ), η E. Λ Λ n A standard fact of finite dimensional diffusion theory (see [14]) is that the differential operator L n generates a Markov process {X n (t)} t in D(R +,E), with transition semigroup {Pt n} t, which is the semigroup of kernel such that P n t F F = t L n P n s Fds for all F D Λ n and t. Step 2. Tightness of finite-volume approximation. We shall prove the tightness of the law of {X n ( )} n 1 in D(R +,E). The tool is Lemma 2.1. Since the configuration space E = M Zd is compact, the compact containedness condition (i) is satisfied automatically. Now we turn to verify the condition (ii). According to Ethier-Kurtz [1, Chapter 3, Theorem 9.4], we only need to show ( T sup E Ln F ( X n (t) ) ) 2 dt < +, T. n 1 This is trivial because (L n F) n 1 are uniformly bounded for each F D. Step 3. Existence of solution to the martingale problem (L, D,ν)in E. By Lemma 2.1, {X n ( )} n 1 is relative compact in law in D(R +,E). By Skorokhod s Lemma, there are a subsequence (n k ) and some probability space (Ω, F, P) such that X n k redefined on (Ω, F, P), converges to X a.s., in the sup-norm topology on D(R +,E). We next to prove that X is a solution to the martingale problem (L, D,ν). For any F D Λ, where Λ Λ N for some N>, and for any n N + R, L n F(η Λn ) = Δ i F(η Λn ) i H Λn (η Λn ), i F(η Λn ) = LF(η). i Λ n i Λ n Since LF is bounded, M t (F ) := F ( X(t) ) F ( X() ) t LF ( X(s) ) ds, as the limit of F ( X n k (t) ) F ( X n k () ) t LF ( X n k (s) ) ds in the sup-norm topology on [,T], for all T>, is a martingale. Hence X is a continuous solution to the martingale problem (L, D, ν).

6 588 L.D. Lemle 3 Proof of the main result At first, we prove a denseness result which is crucial to the proof of Theorem 1.1. Lemma 3.1 Assume (H 1 ) (H 3 ). Then there exists some λ > such that for all λ λ, (λ L)D is dense in (C b (E), ). Proof Since Cb (E) is dense in (C b(e), ), it is enough to show that there exists some λ > such that for each λ λ and for all G Cb (E) such that G(η) = G(η Λ ) for some Λ Z d, there exists a sequence of functions (F n ) n N D such that (λ L)Fn G, as n +. (3.1) Let Λ n =[ n, n] d,n 1. Choose some N such that Λ Λ N. For each n N, consider as before, L n u = Δ i u i H Λn, i u, u D Λn. i Λ n i Λ n By the elliptic PDE theory, for each λ>, λ L n is inversible and F n := (λ L n ) 1 G D Λn D for any n N. Now we claim that F n satisfies (3.1). To this end, let p i := (1 + i ) α,i Z d, where α>dso that i Z d p i <. By Cauchy-Schwarz s inequality, we have (λ L)F n G 2 = (L n L)F n 2 2 = i H i H Λn, i F n i Λ n = pi ( i H i H ), 1 2 Λn i F n pi ( i Λ n i Λn dist(i,λ c n ) R p i Λ:Λ Λ c n i Λ 2) i V Λ i F n 2 pi 1 i Z d (3.2) By the choice of p i, i Λn p i asn +. dist(i,λ c n ) R Λ:Λ Λ c i V Λ 2 is uniformly bounded for all i Z d, by the assumption (H 1 ) (H 3 ) and the compactness i Λ n of the configuration space M Zd. Hence it remains to prove that i Z d if n 2 pi 1 is bounded. Taking gradient on the both sides of following equation (λ L n )F n = G, (3.3)

7 Uniqueness of Fokker-Planck equations for spin lattice systems (I) 589 we have by Weitzenböck-Bochner s formula [15, Theorem 3.33], ( λ Ln + ( Ric i + Hess i,i (H Λn ) )) i F n = i G Hess i,j (H Λn ) j F n. By compactness of the configuration space M Zd, Ric i + Hess i,i (H Λn ) K for some K R. Thus for λ>max{k,},wehave i F n 1 ( ) i G + λ K 1 ( i G + λ K j i j i i j R Hess i,j (H Λn ) j F n j i j i R J(i j) j F n where J(i j):= Λ:i,j Λ Hess i,j (V Λ ). LetJ := sup i Zd J(i). Noticing that i j the cardinality of {j Z d : j i, i j R } is less than (2R + 1) d,wehave and i F n 2 p 1 i Let Since we have (2R + 1) d ( (λ K) 2 i G 2 pi 1 + J 2 j F n j i j i R a(i) := i F n 2 pi 1, b(i):= i G 2 pi 1 p 1 i p j = f(i):= { J 2 (1 + R ) α, if i R ;, otherwise. ) 2 p 1 j ( ) 1 + i α (1 + R ) α, if j i R, 1 + j ( pj p 1 ) ). a(i) (2R + 1) d (λ K) 2 b(i) + (2R + 1) d (f a)(i), (λ K) 2 where f a is the convolution of f and a. For any function g on Z d,let g 1 := i Z d g(i). Then f a 1 = f 1 a 1. For all λ large enough such that we have r := f 1(2R + 1) d (λ K) 2 < 1, a 1 (2R + 1) d (λ K) 2 b 1 + r a 1. i

8 59 L.D. Lemle Consequently a 1 1 (2R + 1) d 1 r (λ K) 2 b 1 <. Therefore (λ L)F n G asn +. The proof is completed. We finish this paper with. Proof of Theorem 1.1 Since every solution to the martingale problem (L, D, ν) is also a solution of Fokker-Planck equation (1.3), the existence has been proved by Theorem 2.2. Next we prove the uniqueness. Let ν t be a weakly continuous solution of Fokker-Planck equation (1.3), satisfying ν t TV ce at for all t and for some constants c 1 and a>, more precisely For each b>a,wehave ν t,f ν,f = t ν s, LF ds, F D. e bt d dt ν t,f dt = e bt ν t, LF dt. (3.4) Integrating by parts, the left hand side of (3.4) equals to then be bt ν t,f dt ν,f, e bt ν t,(b L)F dt = ν,f, F D. (3.5) Let ν t be another weakly continuous solution of Fokker-Planck equation (1.3), ν t TV ce at for all t. Similarly ν t satisfies the equation (3.5). Thus ˆν t := ν t ν t verifies e bt ˆν t,(b L)F dt =, F D. Now by Lemma 3.1, for all b> large enough, i.e., b>b for some constant b >a, (b L)D is dense in (C b (E), ), then e bt ˆν t dt =, b>b. (3.6) By the uniqueness of the Laplace transformation and the continuity of t ˆν t in the weak convergence topology, (3.6) implies ˆν t = for all t. The proof is completed. It would be interesting to prove the uniqueness for equation (1.3) when the spin space M is a non-compact Riemannian manifold. This problem will be studied in a forthcoming paper.

9 Uniqueness of Fokker-Planck equations for spin lattice systems (I) 591 Acknowledgements We are grateful to F.Y. Wang for his remarks. We are particular indebted to the referee for useful comments. L.D. Lemle is partially supported by L.E.A. Math-Mode and People s Republic of China-Romania Joint Research Project no. 4-5/ R. Wang is partially supported by NSFC L. Wu is partially supported by Thousand Talents Program of the Chinese Academy and le projet ANR EVOL. References 1. Albeverio, S., Kondratiev, Y.G., Röckner, M.: Uniqueness of the stochastic dynamics for continuous spin systems on a lattice. J. Funct. Anal. 133, 1 2 (1995) 2. Albeverio, S., Kondratiev, Y.G., Röckner, M.: Dirichlet operators via stochastic analysis. J. Funct. Anal. 128, (1995) 3. Albeverio, S., Röckner, M., Zhang, T.S.: Markov uniqueness and its applications to martingale problems, stochastic differential equations and stochastic quantization. C. R. Math. Rep. Acad. Sci. Can. 15, 1 6 (1993) 4. Bogachev, V.I., Röckner, M., Wang, F.Y.: Invariance implies Gibbsian: some new results. Commun. Math. Phys. 248, (24) 5. Bogachev, V.I., Da Prato, G., Röckner, M.: Fokker-Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces. J. Funct. Anal. 256, (29) 6. Bogachev, V.I., Da Prato, G., Röckner, M.: Existence and uniqueness of solutions for Fokker-Planck equations on Hilbert spaces. J. Evol. Equ. 1, (21) 7. Bogachev, V.I., Da Prato, G., Röckner, M.: Uniqueness for solutions of Fokker-Planck equations on infinite dimensional spaces. Commun. Partial Differ. Equ. 36, (211) 8. Clemens, L.: Ph.D. thesis, MIT, Cambridge, Massachusetts 9. Deuschel, J.D., Stroock, D.W.: Hypercontractivity and spectral gap of symmetric diffusion with applications to the stochastic Ising models. J. Funct. Anal. 92, 3 48 (199) 1. Ethier, N., Kurtz, G.: Markov Processes (Characterization and Convergence). Wiley, New York (1986) 11. Holley, R.A., Stroock, D.W.: A martingale approach to infinite systems of interacting processes. Ann. Probab. 4, (1976) 12. Holley, R.A., Stroock, D.W.: Diffusions on an infinite dimensional torus. J. Funct. Anal. 42, (1981) 13. Holley, R.A., Stroock, D.W.: Logarithmic Sobolev inequalities and stochastic Ising models. J. Stat. Phys. 46, (1987) 14. Ikeda, I., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam (1981) 15. Jost, J.: Reimannian Geometry and Geometric Analysis, 4th edn. Springer, Berlin (25) 16. Röckner, M., Zhang, T.S.: On uniqueness of generalized Schrödinger operators and applications. J. Funct. Anal. 15, (1992) 17. Röckner, M., Zhang, T.S.: On uniqueness of generalized Schrödinger operators, part II. J. Funct. Anal. 119, (1994) 18. Stroock, D.W., Zegarlinski, B.: The equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition. Commun. Math. Phys. 144, (1992) 19. Wu, L., Zhang, Y.: A new topological approach to the L -uniqueness of operators and the L 1 - uniqueness of Fokker-Planck equations. J. Funct. Anal. 241, (26)