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Preprints, Working Papers, ... Year : 2023

EXIT TIME AND PRINCIPAL EIGENVALUE OF NON-REVERSIBLE ELLIPTIC DIFFUSIONS

Abstract

In this work, we analyse the metastability of non-reversible diffusion processes dX t = b(X t)dt + √ h dB t on a bounded domain Ω when b admits the decomposition b = −(∇f + l) and ∇f ⋅l = 0. In this setting, we first show that, when h → 0, the principal eigenvalue of the generator of (X t) t≥0 with Dirichlet boundary conditions on the boundary ∂Ω of Ω is exponentially close to the inverse of the mean exit time from Ω, uniformly in the initial conditions X_0 = x within the compacts of Ω. 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 Ẋ = b(X) admit equilibrium points on ∂Ω. In a second time, when in addition div l = 0, we derive a new sharp asymptotic equivalent in the limit h → 0 of the principal eigenvalue of the generator of the process and of its mean exit time from Ω. 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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Dates and versions

hal-04024094 , version 1 (10-03-2023)

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Dorian Le Peutrec, Laurent Michel, Boris Nectoux. EXIT TIME AND PRINCIPAL EIGENVALUE OF NON-REVERSIBLE ELLIPTIC DIFFUSIONS. 2023. ⟨hal-04024094⟩
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