Stationary Solutions of Stochastic Parabolic and Hyperbolic Sine-Gordon Equations

We 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. We 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→ -∞ we obtain a stationary solution. We show that the stationary solution is localized both in time and space even if the solution at finite time is not.