On continuity equations in infinite dimensions with non-Gaussian reference measure

TL;DR

This paper extends the theory of continuity equations from Gaussian measures to a broad class of non-Gaussian measures in infinite-dimensional spaces, establishing existence and uniqueness of solutions under new conditions.

Contribution

It generalizes recent Gaussian measure results to non-Gaussian measures using measure transportation and logarithmic derivatives, broadening applicability.

Findings

Existence of weak solutions for non-Gaussian measures.

Uniqueness of solutions for a wide class including log-concave Gibbs measures.

Conditions on logarithmic derivatives ensuring well-posedness.

Abstract

Let $\gamma$ be a Gaussian measure on a locally convex space and $H$ be the corresponding Cameron-Martin space. It has been recently shown by L. Ambrosio and A. Figalli that the linear first-order PDE $$ \dot{\rho} + \mbox{div}_{\gamma} (\rho \cdot {b}) =0, \ \ \rho|_{t=0} = \rho_0, $$ where $\rho_0 \cdot \gamma $ is a probability measure, admits a weak solution, in particular, under the following assumptions: $$ \|b\|_{H} \in L^p(\gamma), \ p>1, \ \ \ \exp\bigl(\varepsilon(\mbox{\rm div}_{\gamma} b)_{-} \bigr) \in L^1(\gamma). $$ Applying transportation of measures via triangular maps we prove a similar result for a large class of non-Gaussian probability measures $\nu$ on $\R^{\infty}$, under the main assumption that $\beta_i \in \cap_{n \in \Nat} L^{n}(\nu)$ for every $i \in \Nat$, where $\beta_i$ is the logarithmic derivative of $\nu$ along the coordinate $x_i$. We also show uniqueness of the solution for a wide class of measures. This class includes uniformly log-concave Gibbs measures and certain product measures. measures.