Uniqueness for continuity equations in Hilbert spaces with weakly differentiable drift

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.