On the Smoluchowski-Kramers approximation for a system with infinite degrees of freedom exposed to a magnetic field

TL;DR

This paper investigates the validity of the Smoluchowski-Kramers approximation for a stochastic PDE system with a magnetic field, demonstrating that regularization with friction ensures the approximation's accuracy.

Contribution

It provides a rigorous analysis of the Smoluchowski-Kramers approximation in infinite-dimensional systems with magnetic fields, including justification via regularization.

Findings

The approximation holds when a small friction term is added.

Regularized problems closely approximate the original system.

The study extends the approximation validity to systems with magnetic fields.

Abstract

We study the validity of the so-called Smoluchowski-Kramers approximation for a two dimensional system of stochastic partial differential equations, subject to a constant magnetic field. As the small mass limit does not yield to the solution of the corresponding first order system, we regularize our problem by adding a small friction. We show that in this case the Smoluchowski-Kramers approximation holds. We also give a justification of the regularization, by showing that the regularized problems provide a good approximation to the original ones.