Stationary solutions of stochastic parabolic and hyperbolic sine-Gordon equations

Abstract

The authors consider the damped sine-Gordon equation perturbed by (thermal) space-time noise in the form it arises in the theory of the Josephson junction and charge density waves. They announce a rigorous proof that the coupling constant expansion of the solution of the initial value problem converges for small coupling. Taking the limit of the initial time t0 to - infinity they obtain a stationary solution. It is shown that the stationary solution is localized both in time and space even if the solution at finite time is not.