Uniqueness for continuity equations in Hilbert spaces with weakly differentiable drift

TL;DR

This paper establishes the uniqueness of solutions to continuity equations in infinite-dimensional Hilbert spaces with weakly differentiable drifts, using a novel approach that relaxes previous conditions and employs advanced approximation techniques.

Contribution

It introduces a new method for proving uniqueness that avoids renormalized solutions, utilizing a regularizing Ornstein-Uhlenbeck semigroup and considering Sobolev spaces of vector fields valued in the Hilbert space.

Findings

Proves uniqueness under weaker conditions on the drift's derivative.

Uses a different approximation procedure based on Ornstein-Uhlenbeck semigroup.

Drops exponential integrability conditions on Gaussian divergence, improving existing results.

Abstract

We prove uniqueness for continuity equations in Hilbert spaces $H$. The corresponding drift $F$ is assumed to be in a first order Sobolev space with respect to some Gaussian measure. As in previous work on the subject, the proof is based on commutator estimates which are infinite dimensional analogues to the classical ones due to DiPerna-Lions. Our general approach is, however, quite different since, instead of considering renormalized solutions, we prove a dense range condition implying uniqueness. In addition, compared to known results by Ambrosio-Figalli and Fang-Luo, we use a different approximation procedure, based on a more regularizing Ornstein-Uhlenbeck semigroup and consider Sobolev spaces of vector fields taking values in $H$ rather than the Cameron-Martin space of the Gaussian measure. This leads to different conditions on the derivative of $F$, which are incompatible with previous work on the subject. Furthermore, we can drop the usual exponential integrability conditions on the Gaussian divergence of $F$, thus improving known uniqueness results in this respect.