Absolutely continuous solutions for continuity equations in Hilbert spaces

Abstract

We prove existence of solutions to continuity equations in a separable Hilbert space. We look for solutions which are absolutely continuous with respect to a reference measure γ which is Fomin–differentiable with exponentially integrable partial logarithmic derivatives. We describe a class of examples to which our result applies and for which we can prove also uniqueness. Finally, we consider the case where γ is the invariant measure of a reaction–diffusion equation and prove uniqueness of solutions in this case. We exploit that the gradient operator Dx is closable with respect to Lp(H,γ) and a recent formula for the commutator DxPt−PtDx where Pt is the transition semigroup corresponding to the reaction–diffusion equation, [10]. We stress that Pt is not necessarily symmetric in this case. This uniqueness result is an extension to such γ of that in [12] where γ was the Gaussian invariant measure of a suitable Ornstein–Uhlenbeck process.