Existence and smoothness of the density for the stochastic continuity equation

TL;DR

This paper proves that solutions to the stochastic continuity equation driven by Brownian motion have a smooth density with respect to Lebesgue measure, which is Holder continuous and satisfies Gaussian estimates, using Malliavin calculus.

Contribution

It establishes the existence, smoothness, and regularity properties of the solution's density for the stochastic continuity equation using Malliavin calculus techniques.

Findings

The law of the solution has a density with respect to Lebesgue measure.

The density is Holder continuous.

The density satisfies Gaussian-type estimates.

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 Holder continuous and satisfies some Gaussian-type estimates.