2013/03/28 by Alexander V. Kolesnikov, Michael Röckner, Kolesnikov, Alexander V. +1
Mathematics · #34G10 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:34G10
paper · pdf · doi:10.48550/arxiv.1303.7184
34 pages, minor corrections
arxiv created 2013/12/22 · arxiv updated 2013/12/24
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 PDE ρ + 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(\rm divγ b)- ) ∈ 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 ∈ \Nat Ln(ν) for every i ∈ \Nat, 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. measures.