Exit time and principal eigenvalue of non-reversible elliptic diffusions

Abstract

In this work, we analyse the metastability of non-reversible diffusion processes $$d{X}_t=\mathbf{b}(X_t)dt+\sqrt{h}d{B}_t$$ on a bounded domain $\Omega$ when $\mathbf{b}$ admits the decomposition $\mathbf{b}=-(\nabla f+\mathbf{\ell })$ and $\nabla f\cdot \mathbf{\ell }=0$. In this setting, we first show that, when $h\to 0$, the principal eigenvalue of the generator of $(X_t{)}_{t\ge 0}$ with Dirichlet boundary conditions on the boundary $\partial \Omega$ of $\Omega$ is exponentially close to the inverse of the mean exit time from $\Omega$, uniformly in the initial conditions $X_0=x$ within the compacts of $\Omega$. 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 $\dot{X}=\mathbf{b}(X)$ admit equilibrium points on $\partial \Omega$. In a second time, when in addition $÷\mathbf{\ell }=0$, we derive a new sharp asymptotic equivalent in the limit $h\to 0$ of the principal eigenvalue of the generator of the process and of its mean exit time from $\Omega$. Our proofs combine tools from large deviations theory and from semiclassical analysis, and truly relies on the notion of quasi-stationary distribution.

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