# Existence of a density of the 2Dim Stochastic Navier Stokes Equation driven by Levy processes or fractional Brownian motion

## TL;DR

This paper investigates the regularity of the probability distribution of solutions to the 2D stochastic Navier-Stokes equation driven by Levy processes or fractional Brownian motion, focusing on conditions for absolute continuity of finite-dimensional projections.

## Contribution

It establishes conditions under which the law of the solution's projection is absolutely continuous, advancing understanding of stochastic Navier-Stokes equations with Levy and fractional noise.

## Key findings

- Conditions for absolute continuity of the solution's law
- Regularity properties of the probability measure
- Impact of Levy measure and Hurst parameter

## Abstract

In this article we are interested in the regularity properties of the probability measure induced by the solution process of the L\'evy noise or a fractional Brownian motion driven Navier Stokes Equation on the two dimensional torus $\mathbb{T}$. We mainly investigate under which conditions on the characteristic measure of the L\'evy process or the Hurst parameter of the fractal Brownian motion the law of the projection of $u(t)$ onto any finite dimensional $F\subset L^2(\mathbb{T})$ is absolutely continuous with respect to the Lebesgue measure on $F$.