Sharp asymptotics of metastable transition times for one dimensional SPDEs

Barret, Florent

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 129-166.

Nous nous intéressons à une famille d’équations aux dérivées partielles stochastiques paraboliques et semi-linéaires, perturbées par un bruit blanc en espace-temps, définies sur un intervalle réel compact. Nous cherchons à calculer les asymptotiques précises des espérances des temps de transitions entre les états métastables. Nous démontrons dans ce cadre une version en dimension infinie de la formule dite d’Eyring–Kramers. La preuve repose sur l’approximation par un schéma aux différences finies de l’équation aux dérivées partielles stochastique. L’espérance du temps de transition est calculée pour l’approximation puis contrôlée uniformément quelque soit la dimension.

We consider a class of parabolic semi-linear stochastic partial differential equations driven by space–time white noise on a compact space interval. Our aim is to obtain precise asymptotics of the transition times between metastable states. A version of the so-called Eyring–Kramers formula is proven in an infinite dimensional setting. The proof is based on a spatial finite difference discretization of the stochastic partial differential equation. The expected transition time is computed for the finite dimensional approximation and controlled uniformly in the dimension.

Classification : 82C44, 60H15, 35K57
Mots clés : metastability, metastable transition time, parabolic stochastic partial differential equations, reaction-diffusion equations, stochastic Allen–Cahn equations, Eyring–Kramers formula

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