Absolutely continuous solutions for continuity equations in Hilbert spaces

TL;DR

This paper establishes the existence and uniqueness of absolutely continuous solutions to continuity equations in Hilbert spaces, particularly when the reference measure is invariant for a reaction-diffusion system, extending prior results.

Contribution

It proves existence and uniqueness of solutions in Hilbert spaces with non-Gaussian measures, including invariant measures of reaction-diffusion equations, using advanced operator techniques.

Findings

Existence of solutions under Fomin-differentiability conditions

Uniqueness of solutions for invariant measures of reaction-diffusion equations

Extension of previous Gaussian measure results to broader measures

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 \gamma 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 \gamma is the invariant measure of a reaction-diffusion equation and prove uniqueness of solutions in this case. We exploit that the gradient operator D_x is closable with respect to L^p(H,\gamma) and a recent formula for the commutator D_xP_t - P_tD_x where P_t is the transition semigroup corresponding to the reaction-diffusion equation, [DaDe14]. We stress that P_t is not necessarily symmetric in this case. This uniqueness result is an extension to such \gamma of that in [DaFlRo14] where \gamma was the Gaussian invariant measure of a suitable Ornstein-Uhlenbeck process.