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

Abstract

Let γ 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 PDEρ˙+divγ(ρ⋅b)=0,ρ|t=0=ρ0, where ρ0⋅γ is a probability measure, admits a weak solution, in particular, under the following assumptions:‖b‖H∈Lp(γ),p>1,exp(ε(divγb)−)∈L1(γ). Applying transportation of measures via triangular maps we prove a similar result for a large class of non-Gaussian probability measures ν on R∞, under the main assumption that βi∈⋂n∈NLn(ν) for every i∈N, where βi is the logarithmic derivative of ν along the coordinate xi. 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.