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Temps de transitions métastables pour des systèmes dynamiques stochastiques fini et infini-dimensionnels

Abstract : In this thesis, we work on metastability for some stochastic dynamical systems. More precisely, we study some differential or partial differential equations perturbed by an additive white noise in the small noise asymptotic. We compute the expectation of the transition times for some models (so-called Eyring-Kramers Formula). First we generalize some known results for Itô diffusions whose drift is given by the gradient of a potential. We give an equivalence between the geometry of the potential and an electrical network which allows a simple computation of the transition times between minima of the potential. To do so, we use potential theory and capacities. The main result of this thesis is about a class of scalar, parabolic, semi-linear stochastic partial differential equations perturbed by a space-time white noise on a bounded real interval as the Allen-Cahn model. These equations are similar to the gradient drift diffusions but in infinite dimension. We consider Dirichlet or Neumann boundary conditions and discuss the periodic boundary conditions. Under some assumptions, we prove a formula, similar to the finite dimensional case, for the transition times. In the proof, we use a finite difference approximation and a coupling and apply the finite dimensional estimates to the approximation. We prove the uniformity of the estimates in the dimension and then we take the limit to recover the infinite dimensional equation.
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Contributor : Florent Barret <>
Submitted on : Monday, July 9, 2012 - 11:46:35 AM
Last modification on : Friday, October 23, 2020 - 4:40:13 PM
Long-term archiving on: : Wednesday, October 10, 2012 - 2:30:24 AM


  • HAL Id : tel-00715787, version 1


Florent Barret. Temps de transitions métastables pour des systèmes dynamiques stochastiques fini et infini-dimensionnels. Probabilités [math.PR]. Ecole Polytechnique X, 2012. Français. ⟨tel-00715787⟩



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