Quasi-invariance of Gaussian measures for the periodic Benjamin-Ono-BBM equation

TL;DR

This paper investigates how Gaussian measures evolve under the flow of a Benjamin-Ono type BBM equation with critical dispersion, showing absolute continuity of measures but not their densities.

Contribution

It extends the understanding of measure invariance and transformation properties for a critical dispersion Benjamin-Ono type PDE, highlighting new challenges in identifying densities.

Findings

Gaussian measures become absolutely continuous under the flow

The image measures are supported on fractional Sobolev spaces

The density of the transformed measures remains unidentified

Abstract

The BBM equation is a Hamiltonian PDE which revealed to be a very interesting test-model to study the transformation property of Gaussian measures along the flow. In this paper we study the BBM equation with critical dispersion (which is a Benjamin-Ono type model). We prove that the image of the Gaussian measures supported on fractional Sobolev spaces of increasing regularity are absolutely continuous, but we cannot identify the density, for which new ideas are needed.