Some examples of absolute continuity of measures in stochastic fluid dynamics

TL;DR

This paper investigates the absolute continuity of measures in stochastic fluid dynamics, proving Girsanov theorem applicability and establishing existence, uniqueness, and long-term behavior of solutions for certain stochastic PDEs like Kuramoto-Sivashinsky and Navier-Stokes.

Contribution

It extends Girsanov theorem to specific nonlinear stochastic PDEs and proves existence, uniqueness, and asymptotic properties of their solutions.

Findings

Girsanov theorem holds for 1D stochastic Kuramoto-Sivashinsky equation.

Existence and uniqueness of solutions are established for these stochastic equations.

Asymptotic behavior for large time is characterized.

Abstract

A non linear Ito equation in a Hilbert space is studied by means of Girsanov theorem. We consider a non linearity of polynomial growth in suitable norms, including that of quadratic type which appears in the Kuramoto-Sivashinsky equation and in the Navier-Stokes equation. We prove that Girsanov theorem holds for the 1-dimensional stochastic Kuramoto-Sivashinsky equation and for a modification of the 2- and 3-dimensional stochastic Navier-Stokes equation. In this way, we prove existence and uniqueness of solutions for these stochastic equations. Moreover, the asymptotic behaviour for large time is characterized.