Relations Between Markov Processes Via Local Time and Coordinate Transformations

TL;DR

This paper introduces a modified method using path-dependent transformations to relate different Markov processes, enabling the expression of unknown Green functions in terms of known solutions.

Contribution

It develops a new approach combining time and coordinate transformations to relate Markov processes, extending the DK method to non-Fokker-Planck equations.

Findings

Expressed unknown Green functions via known solutions for specific Markov processes

Extended the DK-transform applicability to a broader class of Markov processes

Provided a practical example with the Schenzle-Brandt process

Abstract

The Duru-Kleinert (DK) method of solving unknown path integrals of quantum mechanical systems by relating them to known ones does not apply to Markov processes since the DK-transform of a Fokker-Planck equation is in general not a Fokker-Planck equation. In this note, we present a significant modification of the method, based again on a combination of path-dependent time and coordinate transformations, to obtain such relations after all. As an application we express unknown Green functions for a one-parameter family of Markov processes in terms of the known one for the Schenzle-Brandt process.