On the convergence to a statistical equilibrium for the Dirac equation

Abstract: We consider the Dirac equation in

R^{3}

with constant coefficients and study the distribution

m u_{t}

of the random solution at time

t i n R

. It is assumed that the initial measure

m u_{0}

has zero mean, a translation-invariant covariance, and finite mean charge density. We also assume that

m u_{0}

satisfies a mixing condition of Rosenblatt- or Ibragimov-Linnik-type. The main result is the convergence of

m u_{t}

to a Gaussian measure as

t o i n f t y

. The proof uses the study of long time asymptotics of the solution and S.N. Bernstein's ``room-corridor method.