The small-mass limit and white-noise limit of an infinite dimensional Generalized Langevin Equation

TL;DR

This paper investigates the asymptotic behavior of the infinite-dimensional Generalized Langevin Equation under small-mass and white-noise limits, demonstrating convergence without requiring Lipschitz conditions on potentials.

Contribution

It introduces a Markovian representation for GLE with power-law decay memory kernels and proves convergence results under minimal assumptions, extending previous analyses.

Findings

Solutions converge in probability in the small-mass limit.

Solutions converge in probability in the white-noise limit.

Under additional regularity, $L^1$ convergence is established.

Abstract

We study asymptotic properties of the Generalized Langevin Equation (GLE) in the presence of a wide class of external potential wells with a power-law decay memory kernel. When the memory can be expressed as a sum of exponentials, a class of Markovian systems in infinite-dimensional spaces is used to represent the GLE. The solutions are shown to converge in probability in the small-mass limit and the white-noise limit to appropriate systems under minimal assumptions, of which no Lipschitz condition is required on the potentials. With further assumptions about space regularity and potentials, we obtain $L^1$ convergence in the white-noise limit.