Generalized Brownian motion from a logical point of view

TL;DR

This paper introduces a logical framework for generalized Brownian motion related to parabolic systems, extending previous constructions and providing a Feynman-Kac formula for probabilistic solutions to complex PDE systems.

Contribution

It offers a new logical construction of generalized Brownian motion for parabolic systems, including coupled second order terms, and derives a Feynman-Kac formula for these processes.

Findings

Constructs generalized Brownian motion using nonstandard analysis.

Derives a Feynman-Kac formula for probabilistic PDE solutions.

Provides tools for designing algorithms for parabolic and elliptic systems.

Abstract

We describe generalized Brownian motion related to parabolic equation systems from a logical point of view, i.e., as a generalization of Anderson's random walk. The connection to classical spaces is based on the Loeb measure. It seems that the construction of Roux in [11] is the only attempt in the literature to define generalized Brownian motion related to parabolic systems with coupled second order terms, where Lam\'e's equation of elastic mechanics is considered as an example. In this paper we provide an exact construction from a logical point of view in a more general situation. A Feynman-Kac formula for generalized Brownian motion is derived which is a useful tool in order to design probabilistic algorithms for Cauchy problems and initial-boundary value (of a class of) parabolic systems as well as for stationary boundary problems of (a class of) elliptic equation systems. The article includes a selfcontained introduction into all tools of nonstandard analysis needed, and which can be read with a minimum knowledge of logic in order to make the results available to a wider audience.