A Pseudo-Differential Operator Construction of Markov Processes Using Feynman Path Integrals

TL;DR

This paper introduces a novel method for constructing Markov processes using pseudo-differential operators with negative definite symbols, representing the fundamental solution as a Feynman path integral, which is a significant departure from traditional approaches.

Contribution

It develops a new framework for Markov process construction via pseudo-differential operators with negative definite symbols and represents solutions as Feynman path integrals.

Findings

Fundamental solutions expressed as Feynman path integrals.

Transition functions represented as pseudo-differential operators with path integral symbols.

Extension of pseudo-differential operator theory to negative definite symbols.

Abstract

In this paper pseudo-differential operators with negative definite symbols are used to construct time- and space-inhomogeneous Markov processes. This is achieved by using the Markov evolution system associated with the fundamental solution of the corresponding pseudo-differential evolution equation. Negative definite symbols are non-standard and differ significantly from the class of H\"{o}rmander type symbols. The novelty of this work is the derivation and the representation of the fundamental solution as a Feynman path integral. This implies that the transition function of the constructed Markov process can be written as a pseudo-differential operator that has a Feynman path integral as its symbol.