Invariant measure for the Klein-Gordon equation in a non periodic setting

TL;DR

This paper constructs an invariant Gibbs measure for the 1D cubic Klein-Gordon equation with a decreasing nonlinearity on the real line, proving almost sure global well-posedness and measure invariance.

Contribution

It introduces a new Gibbs measure for the Klein-Gordon equation with a nonlinearity modulated by an integrable function, establishing invariance and global well-posedness.

Findings

The measure is invariant under the Klein-Gordon flow.

The equation is almost surely globally well-posed in a specific Sobolev space.

The measure construction applies to a non-periodic setting on $\

Abstract

In this paper, we build a Gibbs measure for the 1d cubic Klein-Gordon equation on $\mathbb R$ with a decreasing non linearity, in the sense that the non linearity $f^3$ is multiplied by $\chi$ where $\chi$ is a sufficiently integrable non negative function. We prove that this equation is almost surely globally well-posed in $H^{1/2-}_{\textrm{loc}}$ with respect to this measure and that the measure is invariant under the flow of the equation.