On convergence to equilibrium distribution for wave equation in even dimensions

Abstract

Consider a wave equation (WE) with constant coefficients in $\mathbb{R}^n$ for even $n\ge 2$ and with variable coefficients for even $n\ge4$. We study the distribution $\mu_t$ of the random solution at time $t\in\mathbb{R}$. The initial probability measure $\mu_0$ has a translation-invariant covariance, zero mean and finite mean density for the energy. It also 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 (CLT) for the WE. The proof for the case of constant coefficients is based on stationary phase asymptotics of the solution in the Fourier representation and Bernstein's ‘room–corridor’ argument. The case of variable coefficients is reduced to that of constant ones by a version of the scattering theory, based on Vainberg's results on local energy decay.