Discrete-Time Langevin Motion in a Gibbs Potential

Reza Rastegar,Vadim Roytershteyn,Jiyeon Suh,Alexander Roitershtein

doi:10.4236/am.2012.312a280

Abstract

We consider a multivariate Langevin equation in discrete time, driven by a force induced by certain Gibbs’states. The main goal of the paper is to study the asymptotic behavior of a random walk with stationary increments (which are interpreted as discrete-time speed terms) satisfying the Langevin equation. We observe that (stable) functional limit theorems and laws of iterated logarithm for regular random walks with i.i.d. heavy-tailed increments can be carried over to the motion of the Langevin particle.

Highlights

We start with the following equation describing a discrete-time motion in d, d 1, of a particle with mass m in the presence of a random potential and a viscosity force proportional to velocity: Here d-vector Vn is the velocity at time n, d d matrix represents an anisotropic damping coefficient, and d-vector Fn is a random force applied at time n
The main goal of the paper is to study the asymptotic behavior of a random walk with stationary increments satisfying the Langevin equation
C-chains form an important subclass of chains with complete connections/chains of in-finite order [22,23,24]

Summary

Introduction

We start with the following equation describing a discrete-time motion in d , d 1, of a particle with mass m in the presence of a random potential and a viscosity force proportional to velocity:. Heavy tailed HMM as random coefficients of multivariate linear time-series models have been considered, for instance, in [16,17]. C-chains form an important subclass of chains with complete connections/chains of in-finite order [22,23,24] They can be described as exponentially mixing full shifts, and alternatively defined as an essentially unique random process with a given transition function (g-measure). A3) There exist a constant 0 and a regularly varying function b with index 1 such that for all i D, Q0.i d, ,b with associated measure of regular variation i

Statement of Results

E V I

Proof of Theorem 3

Proof of Theorem 4