Positive Markov processes in Laplace duality

TL;DR

This paper establishes a comprehensive framework for Laplace duality in positive Markov processes, linking duality to complete monotonicity and boundary behaviors, and unifies various process generalizations.

Contribution

It introduces a general Laplace duality framework, characterizes dual processes via complete monotonicity, and refines key duality theorems for positive Markov processes.

Findings

Identifies conditions for Laplace duality in Markov processes.

Provides a unifying structure for generalized branching processes.

Refines Ethier and Kurtz's duality theorem and introduces Laplace symbols.

Abstract

This article develops a general framework for Laplace duality between positive Markov processes in which the one-dimensional Laplace transform of one process can be represented through that of another. We show that a process admits a Laplace dual if and only if it satisfies a certain complete monotonicity condition. Moreover, we analyse how the conventions adopted for the values of $0 \cdot \infty$ and $\infty \cdot 0$ are reflected in the weak continuity/absorptivity properties of the processes in duality at the boundaries $0$ and $\infty$. A broad class of generators admitting Laplace duals is identified, and we provide sufficient conditions under which the associated martingale problems are well-posed with the solutions being in duality at the level of their semigroups. Laplace duality is shown to furnish a unifying structure for several generalizations of continuous-state branching processes, e.g. those with immigration or evolving in random environments. Along the way, a theorem originally due to Ethier and Kurtz -- connecting duality of generators to that of the associated semigroups -- is refined, and we provide a concise proof of the Courr\`ege form for the pointwise infinitesimal generator of a positive Markov process whose domain includes the exponential functions. The latter leads naturally to the notion of a Laplace symbol, which is a parsimonious encoding of the infinitesimal dynamics of the process.