PUB - Publikationen an der Universität Bielefeld

We consider a Kolmogorov operator L-0 in a Hilbert space H, related to a stochastic PDE with a time-dependent singular quasi-dissipative drill 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(vertical bar 0, T vertical bar x H ; nu), p >= 1, where nu(dt, dx) = nu(t)(dx)dt and the family (nu(t))(t is an element of vertical bar 0,T vertical bar) 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. Rockner, Existence of solutions to weak parabolic equations for measures, Proc. London Math. Soc. (3) 88 (2004) 753-774], [V. Bogachev, G. Da Prato, M. Rockner, On parabolic equations for measures, Comm. Partial Differential Equations 33 (3) (2008) 397-418], and [V. Bogachev, G. Da Prato, M. Rockner, W. Stannat, Uniqueness of solutions to weak parabolic equations for measures, Bull. London Math. Soc. 39 (2007) 631-640] to infinite dimensions. (C) 2008 Elsevier Inc. All rights reserved.