The density of the solution to the stochastic transport equation with fractional noise

TL;DR

This paper investigates the stochastic transport equation driven by fractional Brownian motion, establishing existence, uniqueness, and the density's Gaussian bounds for its weak solution using Malliavin calculus.

Contribution

It provides new results on the existence, uniqueness, and density estimates for solutions to stochastic transport equations with fractional noise.

Findings

Existence and uniqueness of the weak solution.

Proven the existence of the solution's density.

Derived Gaussian upper and lower bounds for the density.

Abstract

We consider the transport equation driven by the fractional Brownian motion. We study the existence and the uniqueness of the weak solution and, by using the tools of the Malliavin calculus, we prove the existence of the density of the solution and we give Gaussian estimates from above and from below for this density.