Rare Events and Exit Problems for Stochastic Equations: Theory and Numerics

Abstract

This dissertation is concerned with the small-noise asymptotics of stochastic differential equations and stochastic partial differential equations. In the first part of the manuscript, we present an overview of large deviations theory in the context of stochastic differential equations, with a particular focus on describing the long time behavior of the system in the presence of point attractors. We then present results describing a novel algorithm for computing the \textit{quasi-potential}, a key quantity in large deviations theory, for two-dimensional stochastic differential equations. Our solver, the Efficient Jet Marcher, computes the quasi-potential $U(x)$ on a mesh by propagating $U$ and $\nabla U$ outward away from the attractor. By using higher-order interpolation schemes and approximations for the minimum action paths, we are able to achieve 2nd order accuracy in the mesh spacing $h$.

In the second part of the manuscript, we consider two important problems in large deviations theory for stochastic partial differentiable equations. First, we consider stochastic reaction-diffusion equations posed on a bounded domain, which contain both a large diffusion term and a small noise term. We prove that in the joint small noise and large diffusion limits, the system satisfies a large deviations principle with respect to an action functional that is finite only on paths that are constant in the spatial variable. We then use this result to compute asymptotics of the first exit time of the solution from bounded domains in function spaces. Second, we consider the two-dimensional stochastic Navier-Stokes equations posed on the torus. In the simultaneous limit as the noise magnitude and noise regularization are both sent to $0$, the solutions converge to the deterministic Navier-Stokes equations. We prove that the invariant measures, which converge to a Dirac mass at $0$, also satisfy a large deviation principle with action functional given by the enstrophy.