The first exit problem of reaction-diffusion equations for small\n multiplicative L 'evy noise

2017/06/23 by Michael A. Högele, Högele, Michael A.
Computer Science · Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #35K05 #35K55 #35K57 #35K91 #37D15 #37L55 #60G51 #60G52 #60G55 #60H15 #Advanced Mathematical Modeling in Engineering #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1706.07745

openalex publication_date 2017/06/23 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

This article studies the dynamics of a nonlinear dissipative\nreaction-diffusion equation with well-separated stable states which is\nperturbed by infinite-dimensional multiplicative L 'evy noise with a regularly\nvarying component at intensity \ε>0. The main results establish the\nprecise asymptotics of the first exit times and locus of the solution\nX^\ε from the domain of attraction of a deterministic stable state, in\nthe limit as \ε\→ 0. In contrast to the exponential growth for\nrespective Gaussian perturbations the exit times grow essentially as a power\nfunction of the noise intensity as \ε \→ 0 with the exponent\ngiven as the tail index -\α, \α>0, of the L 'evy measure,\nanalogously to the case of additive noise in Debussche et al (2013). In this\narticle we substantially improve their quadratic estimate of the small jump\ndynamics and derive a new exponential estimate of the stochastic convolution\nfor stochastic L 'evy integrals with bounded jumps based on the recent pathwise\nBurkholder-Davis-Gundy inequality by Siorpaes (2018). This allows to cover\nperturbations with general tail index \α>0, multiplicative noise and\nperturbations of the linear heat equation. In addition, our convergence results\nare probabilistically strongest possible. Finally, we infer the metastable\nconvergence of the process on the common time scale t/\ε^\α to a\nMarkov chain switching between the stable states of the deterministic dynamical\nsystem.\n

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