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

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, after [27].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.

Quasi-invariance of Gaussian measures of negative regularity for fractional nonlinear Schrödinger equations

“…Indeed, many results have appeared regarding the quasiinvariance of Gaussian measures for various different dispersive PDEs. In particular, there are results for quasi-invariance of the BBM and Benjamin-Ono equations [81,32,33], KdV type equations [72], wave equations [66,36,74], and Schrödinger equations [65,62,64,71,29,19,60,30,32]. The key underlying observation in this study is that the quasi-invariance of Gaussian measures is intimately tied to the dispersive character of the equation; see [62,74] for negative results for some dispersionless ODEs.…”