Smoluchowski-Kramers approximation and large deviations for infinite dimensional gradient systems

TL;DR

This paper computes quasi-potentials for a stochastic wave equation of gradient type, demonstrating that the Smoluchowski-Kramers approximation effectively captures long-term behavior in the zero noise limit, independent of mass density.

Contribution

It explicitly calculates quasi-potentials for the stochastic wave equation and shows their relation to the heat equation's quasi-potential, validating the approximation for long-term dynamics.

Findings

Quasi-potentials are independent of mass density in the gradient case.

The quasi-potential matches that of the stochastic heat equation in the zero mass limit.

Smoluchowski-Kramers approximation accurately predicts exit times and locations.

Abstract

In this paper, we explicitly calculate the quasi-potentials for the damped semilinear stochastic wave equation when the system is of gradient type. We show that in this case the infimum of the quasi-potential with respect to all possible velocities does not depend on the density of the mass and does coincide with 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 can be used to approximate long time behavior in the zero noise limit, such as exit time and exit place from a basin of attraction.