Stationary state solutions for a gently stochastic nonlinear wave equation with ultraviolet cutoffs

Abstract

We consider a non-linear, one-dimensional wave equation system with finite-dimensional stochastic driving terms and with weak dissipation. A stationary process that solves the system is used to model steady-state non-equilibrium heat flow through a non-linear medium. We show existence and uniqueness of invariant measures for the system modified with ultraviolet cutoffs, and we obtain estimates for the field covariances with respect to these measures, estimates that are uniform in the cutoffs. Finally, we discuss the limit of these measures as the ultraviolet cutoffs are removed.