Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation

TL;DR

This paper constructs weighted Gaussian measures linked to higher order conservation laws of the Benjamin-Ono equation, supported by Sobolev spaces, and explores their potential invariance under the equation's flow.

Contribution

It introduces a novel construction of Gaussian measures for the Benjamin-Ono equation's conservation laws, extending previous work on similar equations.

Findings

Measures supported in Sobolev spaces of increasing regularity

Support property suggesting invariance under Benjamin-Ono flow

Conjecture on measure invariance inspired by analogous results

Abstract

Inspired by the work of Zhidkov on the KdV equation, we perform a construction of weighted gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation. The resulting measures are supported by Sobolev spaces of increasing regularity. We also prove a property on the support of these measures leading to the conjecture that they are indeed invariant by the flow of the Benjamin-Ono equation.