Asymptotic analysis of exit time for dynamical systems with a single well potential

TL;DR

This paper provides a detailed asymptotic analysis of the expected exit time for a stochastic dynamical system with a single potential well, including explicit formulas for the dominant exponential term and power series corrections.

Contribution

It introduces a complete asymptotic expansion for the exit time function and eigenvalues in systems with a single potential well, accounting for degenerate minima and boundary conditions.

Findings

Explicit exponential term for exit time derived

Power series expansion constructed for accuracy

Asymptotic formulas applicable to degenerate minima

Abstract

We study the exit time from a bounded multi-dimensional domain $\Omega$ of the stochastic process $\mathbf{Y}_\varepsilon=\mathbf{Y}_\varepsilon(t,a)$, $t\geqslant 0$, $a\in \mathcal{A}$, governed by the overdamped Langevin dynamics \begin{equation*} d\mathbf{Y}_\varepsilon =-\nabla V(\mathbf{Y}_\varepsilon) dt +\sqrt{2}\varepsilon\, d\mathbf{W}, \qquad \mathbf{Y}_\varepsilon(0,a)\equiv x\in\Omega \end{equation*} where $\varepsilon$ is a small positive parameter, $\mathcal{A}$ is a sample space, $\mathbf{W}$ is a $n$-dimensional Wiener process. The exit time corresponds to the first hitting of $\partial\Omega$ by the trajectories of the above dynamical system and the expectation value of this exit time solves the boundary value problem \begin{equation*} (-\varepsilon^2\Delta +\nabla V\cdot \nabla)u_\varepsilon=1\quad\text{in}\quad\Omega,\qquad u_\varepsilon=0\quad\text{on}\quad\partial\Omega. \end{equation*} We assume that the function $V$ is smooth enough and has the only minimum at the origin (contained in $\Omega$); the minimum can be degenerate. At other points of $\Omega$, the gradient of $V$ is non-zero and the normal derivative of $V$ at the boundary $\partial\Omega$ does not vanish as well. Our main result is a complete asymptotic expansion for $u_\varepsilon$ as well as for the lowest eigenvalue of the considered problem and for the associated eigenfunction. The asymptotics for $u_\varepsilon$ involves a term exponentially large $\varepsilon$; we find this term in a closed form. Apart of this term, we also construct a power in $\varepsilon$ asymptotic expansion such that this expansion and a mentioned exponentially large term approximate $u_\varepsilon$ up to arbitrarily power of $\varepsilon$. We also discuss some probabilistic aspects of our results.