Canonical Sample Spaces for Random Dynamical Systems

TL;DR

This paper reviews the construction of canonical sample spaces for various types of noises in stochastic differential equations, emphasizing their mathematical properties and relevance for dynamical systems analysis.

Contribution

It introduces and analyzes canonical sample spaces with metric and topological structures for different noises, facilitating a dynamical systems approach to stochastic differential equations.

Findings

Canonical sample spaces are constructed for diverse noises.

These spaces have properties like separability and completeness.

They include a shift flow with an invariant measure.

Abstract

This is an overview about natural sample spaces for differential equations driven by various noises. Appropriate sample spaces are needed in order to facilitate a random dynamical systems approach for stochastic differential equations. The noise could be white or colored, Gaussian or non-Gaussian, Markov or non-Markov, and semimartingale or non-semimartingale. Typical noises are defined in terms of Brownian motion, Levy motion and fractional Brownian motion. In each of these cases, a canonical sample space with an appropriate metric (or topology that gives convergence concept) is introduced. Basic properties of canonical sample spaces, such as separability and completeness, are then discussed. Moreover, a flow defined by shifts, is introduced on these canonical sample spaces. This flow has an invariant measure which is the probability distribution for Brownian motion, or Levy motion or fractional Brownian motion. Thus canonical sample spaces are much richer in mathematical structures than the usual sample spaces in probability theory, as they have metric or topological structures, together with a shift flow (or driving flow) defined on it. This facilitates dynamical systems approaches for studying stochastic differential equations.