Uniform attraction and exit problems for stochastic damped wave equations

TL;DR

This paper studies stochastic damped wave equations with polynomial nonlinearities, analyzing deterministic stability, exit times, and places under small noise, and introduces new controllability and boundary concepts for infinite-dimensional systems.

Contribution

It develops novel large deviations and controllability techniques for stochastic damped wave equations, including a new notion of regular boundary points in infinite dimensions.

Findings

Uniform attraction neighborhoods are established for deterministic dynamics.

Logarithmic asymptotics for exit times and places are derived in the small noise limit.

Lower bounds for mean explosion times of solutions are provided.

Abstract

We consider a class of wave equations with constant damping and polynomial nonlinearities that are perturbed by small, multiplicative, space-time white noise. The equations are defined on a one-dimensional bounded interval with Dirichlet boundary conditions, continuous initial position and distributional initial velocity. In the first part of this work, we study the corresponding deterministic dynamics and prove that certain neighborhoods of asymptotically stable equilibria are uniformly attracting in the topology of uniform convergence. Then, we consider exit problems for local solutions of the stochastic damped wave equations from bounded domains $D$ of uniform attraction. Using tools from large deviations along with novel controllability results, we obtain logarithmic asymptotics for exit times and exit places, in the vanishing noise limit, that are expressed in terms of the corresponding quasipotential. In doing so, we develop arguments that take into account the lack of both smoothing and exact controllability that are inherent to the problem at hand. Moreover, our exit time results provide asymptotic lower bounds for the mean explosion time of local solutions. We introduce a novel notion of "regular" boundary points allowing to avoid the question of boundary smoothness in infinite dimensions and leading to the proof of a large deviations lower bound for the exit place. We illustrate this notion by providing explicit examples for different classes of domains $D$. Conditions under which lower and upper bounds for exit time and exit place logarithmic asymptotic hold, are also presented. In addition, we obtain deterministic stability results for linear damped wave equations that are of independent interest.