L(R, dx)-UNIQUENESS OF WEAK SOLUTIONS FOR THE FOKKER-PLANCK EQUATION ASSOCIATED WITH A CLASS OF DIRICHLET OPERATORS

. The main purpose of this paper is to prove the ( L 1 ( R d ,dx ) , k . k 1 )- uniqueness of the Fokker-Planck equation associated with the non-symmetric Dirichlet operator

Abstract. The main purpose of this paper is to prove the (L 1 (R d , dx), . 1 )uniqueness of the Fokker-Planck equation associated with the non-symmetric Dirichlet operator where b : R d → R d is a measurable locally bounded vector field and V : R d → R is a locally bounded non-negative potential.

Preliminaries
In general, for a C 0 -semigroup {T (t)} t≥0 on L 1 (R d , dx), its adjoint semigroup {T * (t)} t≥0 is no longer strongly continuous on the dual topological space L ∞ (R d , dx) of L 1 (R d , dx) with respect to the strong dual topology of L ∞ (R d , dx).
Recentely Wu and Zhang [21] introduced on L ∞ (R d , dx) the topology of uniform convergence on compact subsets of (L 1 (R d , dx), . 1 ), denoted by C(L ∞ , L 1 ). If In accord with the uniqueness notion used by Arendt [3], Eberle [8], Röckner [15], Wu [19] and [20], Arendt, Metafune and Pallara [4], Wu and Zhang [21] and others, we can introduce L ∞ (R d , dx), C L ∞ , L 1 -uniqueness of pre-generators. We say that a pre-generator A is L ∞ (R d , dx), C L ∞ , L 1 -unique if A is closable and its closure A with respect to the topology C L ∞ , L 1 is the generator of some C 0 -semigroup on L ∞ (R d , dx), C L ∞ , L 1 .

LUDOVIC DAN LEMLE
The main result concerning L ∞ (R d , dx), C L ∞ , L 1 -uniqueness of pre-generators is (see [21] or [10] for much more general results) The following assertions are equivalent: (iv) (uniqueness of weak solutions for the dual Cauchy problem) for every func- We remark that if A is a second order elliptic differential operator with domain D = C ∞ 0 (R d ), the space of infinitely differentiable functions with compact support in R d , then the weak solutions for the dual Cauchy problem in part (iv) of Theorem 1.1 correspond exactly to those in the distribution sense in the theory of partial differential equations and the dual Cauchy problem becomes the Fokker-Planck equation for heat diffusion in the sense of distributions.
The (L 1 (R d , dx), . 1 )-uniqueness of weak solutions of the Fokker-Planck equation is very important from the point of view of heat diffusion. Usually the energy in a system at time t is considered to be In this paper we consider the diffusion operator where b : R d → R d is a measurable locally bounded vector field and V : R d → R is a locally bounded potential (here · denotes the inner product in R d ).
The study of this operator has attracted much attention both from the people working on Nelson's stochastic mechanics and from those working on the theory of Dirichlet forms. We are content here only to cite the papers of Meyer and Zheng [14], Carmona [7], Albeverio, Brasche and Röckner [1], Albeverio, Kondratiev and Röckner [2], Bogacev, Krylov and Röckner [5], Manca [13], Bogacev, Da Prato, Röckner and Stannat [6], where the reader can find a large number of references.
Our main purpose in this paper is to prove the (L 1 (R d , dx), . 1 )-uniqueness of the Fokker-Planck equation associated with the non-symmetric Dirichlet operator For this purpose we use the equivalence between the (L 1 (R d , dx), . 1 )-uniqueness of the Fokker-Planck equation associated with A V and the L ∞ (R d , dx), C L ∞ , L 1 -uniqueness of A V in Theorem 1.1 (see [11] for detailed proofs).

The main results
At first, we must remark that the Dirichlet operator (A V , C ∞ 0 (R d )) is a pregenerator on L ∞ (R d , dx). Indeed, if we consider the Feynman-Kac semigroup P V t t≥0 given by where (X t ) 0≤t<τe is the diffusion generated by A and τ e is the explosion time, then by [21, Theorem 1.4] P V t t≥0 is a C 0 -semigroup on L ∞ (R d , dx) with respect to the topology C(L ∞ , L 1 ). Let ∂ be the point at infinity of R d . If we put X t = ∂ after the explosion time t ≥ τ e , then by Ito's formula it follows for any f ∈ C ∞ 0 (R d ) that is a local martingale. As it is bounded over bounded times intervals, it is a true martingale. Thus by taking the expectation under P x , we get Therefore f belongs to the domain of the generator is a pre-generator on L ∞ (R d , dx) and we can apply Theorem 1.1 to study the (L 1 (R d , dx), . 1 )uniqueness of weak solutions of the Fokker-Planck equation associated with this operator.
Denote by |x| = √ x · x the Euclidean norm in R d . If there is some mesurable locally bounded function β : then for any initial point x = 0 we have In other words, |X t | goes to infinity more rapidly than the one-dimensional diffusion generated by This is standard in probability (see Ikeda and Watanabe [9]). We remark that for f ∈ C ∞ 0 (0, ∞), the one-dimensional operator can be written in the Feller form In particular, for V = 0, the one-dimensional operator If the one-dimensional diffusion operator is L ∞ (0, ∞; ρdx)-unique with respect to the topology C(L ∞ (0, ∞; ρdx), L 1 (0, ∞; ρdx)), then for any f ∈ L 1 (R d , dx) the Fokker-Planck equation ∂ t u(t, x) = 1 2 ∆u(t, x) − div (b(x)u(t, x)) − V (x)u(t, x) u(0, x) = f (x) has a L 1 (R d , dx)-unique weak solution which is given by u(t, x) = P V t f (x).