Uniform attraction and exit problems for stochastic damped wave equations
Ioannis Gasteratos, Michael Salins, Konstantinos Spiliopoulos
TL;DR
This work develops a Freidlin–Wentzell-type exit theory for small-noise stochastic damped wave equations with polynomial nonlinearities in one dimension. By working in the uniform topology $\mathcal{E}=C_0(0,\ell)\oplus C^{-1}(0,\ell)$, the authors prove uniform attraction to stable equilibria for the deterministic dynamics and establish local uniform large deviations for local solutions, despite lack of smoothing and exact controllability. They introduce controllability-based quasipotentials and a notion of regular boundary points to obtain sharp upper and lower bounds for exit times and exit locations, including explicit exit-shape asymptotics. Deterministic linear stability results and global-wellposedness in $\mathcal{E}$ under stronger growth assumptions round out the analysis. The results provide quantitative, exponentially accurate descriptions of metastable transitions and mean explosion times for local solutions in a challenging infinite-dimensional, non-smoothing setting.
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.