The Annals of Probability

Smoluchowski–Kramers approximation and large deviations for infinite-dimensional nongradient systems with applications to the exit problem

Sandra Cerrai and Michael Salins

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Abstract

In this paper, we study the quasi-potential for a general class of damped semilinear stochastic wave equations. We show that as the density of the mass converges to zero, the infimum of the quasi-potential with respect to all possible velocities converges to the quasi-potential of the corresponding stochastic heat equation, that one obtains from the zero mass limit. This shows in particular that the Smoluchowski–Kramers approximation is not only valid for small time, but in the zero noise limit regime, can be used to approximate long-time behaviors such as exit time and exit place from a basin of attraction.

Article information

Source
Ann. Probab., Volume 44, Number 4 (2016), 2591-2642.

Dates
Received: March 2014
Revised: April 2015
First available in Project Euclid: 2 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1470139150

Digital Object Identifier
doi:10.1214/15-AOP1029

Mathematical Reviews number (MathSciNet)
MR3531676

Zentralblatt MATH identifier
1350.60054

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60F10: Large deviations 35K57: Reaction-diffusion equations 49J45: Methods involving semicontinuity and convergence; relaxation

Keywords
Stochastic wave equations stochastic parabolic equations singular perturbations large deviations exit problem

Citation

Cerrai, Sandra; Salins, Michael. Smoluchowski–Kramers approximation and large deviations for infinite-dimensional nongradient systems with applications to the exit problem. Ann. Probab. 44 (2016), no. 4, 2591--2642. doi:10.1214/15-AOP1029. https://projecteuclid.org/euclid.aop/1470139150


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