Existence and Smoothness of the Density for the Stochastic Continuity Equation

Abstract

We consider the stochastic continuity equation driven by Brownian motion. We use the techniques of the Malliavin calculus to show that the law of the solution has a density with respect to the Lebesgue measure. We also prove that the density is Hölder continuous and satisfies some Gaussian-type estimates.MSC 2010 : Primary 60F05: Secondary 60H05, 91G70.