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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)\equivx\in\Omega \end{equation*} where $\varepsilon$ is a small positive parameter,$\mathcal{A}$ is a sample space, $\mathbf{W}$ is a $n$-dimensional Wienerprocess. The exit time corresponds to the first hitting of $\partial\Omega$ bythe trajectories of the above dynamical system and the expectation value ofthis exit time solves the boundary value problem \begin{equation*}(-\varepsilon^2\Delta +\nabla V\cdot\nabla)u_\varepsilon=1\quad\text{in}\quad\Omega,\qquadu_\varepsilon=0\quad\text{on}\quad\partial\Omega. \end{equation*} We assumethat 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 $V$ atthe boundary $\partial\Omega$ does not vanish as well. Our main result is acomplete asymptotic expansion for $u_\varepsilon$ as well as for the lowesteigenvalue of the considered problem and for the associated eigenfunction. Theasymptotics for $u_\varepsilon$ involves a term exponentially large$\varepsilon$; we find this term in a closed form. Apart of this term, we alsoconstruct a power in $\varepsilon$ asymptotic expansion such that thisexpansion and a mentioned exponentially large term approximate $u_\varepsilon$up to arbitrarily power of $\varepsilon$. We also discuss some probabilisticaspects of our results.

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