{ "id": "1906.04715", "version": "v1", "published": "2019-06-11T17:37:28.000Z", "updated": "2019-06-11T17:37:28.000Z", "title": "Asymptotic analysis of exit time for dynamical systems with a single well potential", "authors": [ "D. Borisov", "O. Sultanov" ], "categories": [ "math.AP", "math-ph", "math.MP", "math.PR" ], "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.", "revisions": [ { "version": "v1", "updated": "2019-06-11T17:37:28.000Z" } ], "analyses": { "subjects": [ "35B25", "35C20" ], "keywords": [ "dynamical system", "asymptotic analysis", "mentioned exponentially large term approximate", "complete asymptotic expansion", "boundary value problem" ], "note": { "typesetting": "TeX", "pages": 0, "language": "en", "license": "arXiv", "status": "editable" } } }