On Convergence to Equilibrium Distribution for Dirac Equation

#### A.I. Komech, E.A. Kopylova

2011, v.17, Issue 4, 523-540

ABSTRACT

We consider the Dirac equation in $\r^3$ with a potential, and study the distribution $\mu_t$ of the random solution at time $t\in\r$. The initial measure $\mu_0$ has zero mean, a translation-invariant covariance, and a finite mean charge density. We also assume that $\mu_0$ satisfies a mixing condition of Rosenblatt- or Ibragimov - Linnik-type. The main result is the long time convergence of projection of $\mu_t$ onto the continuous spectral space. The limiting measure is Gaussian.

Keywords: Dirac equation,random initial data,mixing condition,Gaussianmeasures,covariance matrices,characteristic functional,scattering theory