Fokker–Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces

Abstract

We consider a Kolmogorov operator L 0 in a Hilbert space H, related to a stochastic PDE with a time-dependent singular quasi-dissipative drift F = F ( t , ⋅ ) : H → H , defined on a suitable space of regular functions. We show that L 0 is essentially m-dissipative in the space L p ( [ 0 , T ] × H ; ν ) , p ⩾ 1 , where ν ( d t , d x ) = ν t ( d x ) d t and the family ( ν t ) t ∈ [ 0 , T ] is a solution of the Fokker–Planck equation given by L 0 . As a consequence, the closure of L 0 generates a Markov C 0 -semigroup. We also prove uniqueness of solutions to the Fokker–Planck equation for singular drifts F. Applications to reaction–diffusion equations with time-dependent reaction term are presented. This result is a generalization of the finite-dimensional case considered in [V. Bogachev, G. Da Prato, M. Röckner, Existence of solutions to weak parabolic equations for measures, Proc. London Math. Soc. (3) 88 (2004) 753–774], [V. Bogachev, G. Da Prato, M. Röckner, On parabolic equations for measures, Comm. Partial Differential Equations 33 (3) (2008) 397–418], and [V. Bogachev, G. Da Prato, M. Röckner, W. Stannat, Uniqueness of solutions to weak parabolic equations for measures, Bull. London Math. Soc. 39 (2007) 631–640] to infinite dimensions.