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

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 $$\dot{\rho }+di{v}_{\gamma }(\rho \cdot b)=0,\rho {\mathrm{\mid }}_{t=0}={\rho }_0,$$ where ${\rho }_0\cdot \gamma$ is a probability measure, admits a weak solution, in particular, under the following assumptions: $$\mathrm{\parallel }b{\mathrm{\parallel }}_H\in L^p(\gamma ),p>1,\exp (\epsilon (\mathrm{d}\mathrm{i}{\mathrm{v}}_{\gamma }\mathrm{b}{)}_{-})\in {\mathrm{L}}^1(\gamma )\mathrm{.}$$ Applying transportation of measures via triangular maps we prove a similar result for a large class of non-Gaussian probability measures $\nu$ on ${\mathbb{R}}^{\mathrm{\infty }}$, under the main assumption that $\beta_i \in \cap_{n \in \Nat} L^{n}(\nu)$forevery$i \in \Nat$,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.