Exit Time and Principal Eigenvalue of Non-reversible Elliptic Diffusions

In this work, we analyse the metastability of non-reversible diffusion processes \(\begin{aligned} dX_t=\varvec{b}(X_t)dt+\sqrt{h} \, dB_t \end{aligned}\) $$\begin{array}{c}\hfill d{X}_t=\mathit{b}(X_t)dt+\sqrt{h}d{B}_t\end{array}$$ on a bounded domain \(\Omega \) $\mathrm{\Omega }$ when \(\varvec{b}\) $\mathit{b}$ admits the decomposition \(\varvec{b}=-(\nabla f+\varvec{\ell })\) $\mathit{b}=-(\mathrm{\nabla }f+\mathit{\ell })$ and \(\nabla f \cdot \varvec{\ell }=0\) $\mathrm{\nabla }f·\mathit{\ell }=0$ . In this setting, we first show that, when \(h\rightarrow 0\) $h\to 0$ , the principal eigenvalue of the generator of \((X_t)_{t\ge 0}\) ${(X_t)}_{t\ge 0}$ with Dirichlet boundary conditions on the boundary \(\partial \Omega \) $\partial \mathrm{\Omega }$ of \(\Omega \) $\mathrm{\Omega }$ is exponentially close to the inverse of the mean exit time from \(\Omega \) $\mathrm{\Omega }$ , uniformly in the initial conditions \(X_0=x\) $X_0=x$ within the compacts of  \(\Omega \) $\mathrm{\Omega }$ . The asymptotic behavior of the law of the exit time in this limit is also obtained. The main novelty of these first results follows from the consideration of non-reversible elliptic diffusions whose associated dynamical systems \(\dot{X}=\varvec{b}(X)\) $\dot{X}=\mathit{b}(X)$ admit equilibrium points on \(\partial \Omega \) $\partial \mathrm{\Omega }$ . In a second time, when in addition \({\text {div}}\varvec{\ell } =0\) $\text{div}\mathit{\ell }=0$ , we derive a new sharp asymptotic equivalent in the limit  \(h\rightarrow 0\) $h\to 0$ of the principal eigenvalue of the generator of the process and of its mean exit time from \(\Omega \) $\mathrm{\Omega }$ . Our proofs combine tools from large deviations theory and from semiclassical analysis, and heavily rely on the notion of quasi-stationary distribution.