Invariant measure and large time dynamics of the cubic Klein-Gordon equation in $3D$

TL;DR

This paper constructs an invariant probability measure for the cubic Klein-Gordon equation in 3D, enabling analysis of its long-term probabilistic dynamics and extending the FDL approach to PDEs with a single conservation law.

Contribution

It introduces a new invariant measure for the 3D cubic Klein-Gordon equation, including wave equations, and explores its implications for long-term behavior and measure properties.

Findings

Invariant measure concentrated on H^2 x H^1 for the equation.

Results on the long-time probabilistic dynamics of the flow.

Extension of FDL approach to PDEs with one conservation law.

Abstract

In this paper we construct an invariant probability measure concentrated on $H^2(K)\times H^1(K)$ for a general cubic Klein-Gordon equation (including the case of the wave equation). Here $K$ represents both the $3$-dimensional torus or a bounded domain with smooth boundary in $\Bbb R^3.$ That allows to deduce some corollaries on the long time behaviour of the flow of the equation in a probabilistic sense. We also establish qualitative properties of the constructed measure. This work extends the Fluctuation-Dissipation-Limit (FDL) approach to PDEs having only one (coercive) conservation law.