On the dual representations of Laplace transforms of Markov processes

TL;DR

This paper introduces a comprehensive framework for dual representations of Laplace transforms of Markov processes, enabling new insights into their structure and applications in stochastic models.

Contribution

It generalizes existing dual representation methods, encompassing recent examples and extending to new classes of processes like Lévy and birth-death processes.

Findings

Unified framework for dual Laplace transform representations

Includes new examples involving Lévy and birth-death processes

Facilitates analysis of stochastic models like ASEP and KPZ

Abstract

We provide a general framework for dual representations of Laplace transforms of Markov processes. Such representations state that the Laplace transform of a finite-dimensional distribution of a Markov process can be expressed in terms of a Laplace transform involving another Markov process, but with coefficients in the Laplace transform and time indices of the process interchanged. Dual representations of Laplace transforms have been used recently to study open ASEP and to describe stationary measures of the open KPZ equation. Our framework covers both recently discovered examples in the literature and several new ones, involving general L\'evy processes and certain birth-and-death processes.