On Convergence to Equilibrium Distribution, I. The Klein - Gordon Equation with Mixing

TL;DR

This paper proves that solutions to the Klein-Gordon equation with random initial data converge to a Gaussian distribution over time, establishing a Central Limit Theorem for such solutions in both constant and variable coefficient cases.

Contribution

It demonstrates the convergence of the distribution of solutions to a Gaussian, extending CLT results to the Klein-Gordon equation with mixing initial measures.

Findings

Solutions' distributions converge to Gaussian as time approaches infinity.

Established CLT for Klein-Gordon equation with mixing initial data.

Extended analysis to variable coefficient cases using scattering theory.

Abstract

Consider the Klein-Gordon equation (KGE) in $\R^n$, $n\ge 2$, with constant or variable coefficients. We study the distribution $\mu_t$ of the random solution at time $t\in\R$. We assume that the initial probability measure $\mu_0$ has zero mean, a translation-invariant covariance, and a finite mean energy density. We also asume that $\mu_0$ satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. The main result is the convergence of $\mu_t$ to a Gaussian probability measure as $t\to\infty$ which gives a Central Limit Theorem for the KGE. The proof for the case of constant coefficients is based on an analysis of long time asymptotics of the solution in the Fourier representation and Bernstein's `room-corridor' argument. The case of variable coefficients is treated by using an `averaged' version of the scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay.