Stochastic Manifestations of Chaos

Abstract

The dynamics of a large class of stochastic systems is governed by Hamiltonians that can be obtained from the stochastic dynamical equations of these systems via an appropriate transformation, [Risken 1984], [Graham and Tel 1984], [Reichl 1998]. When the Hamiltonian system exhibits a transition to chaos, the decay rates of the stochastic equations show level repulsion. In the sections below, we demonstrate this correspondence between stochastic systems and deterministic Hamiltonian systems, for three different types of stochastic processes. We first consider Brownian motion in two space dimensions, then a random walk in two space dimensions, and finally a Brownian motion in one space dimension, driven by a time-periodic force.