Reciprocal processes. A measure-theoretical point of view

TL;DR

This survey paper explores reciprocal processes from a measure-theoretical perspective, highlighting their properties, structures, and differences from Markov processes, with applications to diffusion and jump processes.

Contribution

It provides a unified measure-theoretical framework for understanding reciprocal processes, including diffusion and jump processes, and clarifies their properties and time-symmetries.

Findings

Reciprocal processes generalize Markov bridges with unique properties.

The measure-theoretical approach unifies diffusion and jump process analysis.

Examples illustrate the properties and distinctions of reciprocal processes.

Abstract

This is a survey paper about reciprocal processes. The bridges of a Markov process are also Markov. But an arbitrary mixture of these bridges fails to be Markov in general. However, it still enjoys the interesting properties of a reciprocal process. The structures of Markov and reciprocal processes are recalled with emphasis on their time-symmetries. A review of the main properties of the reciprocal processes is presented. Our measure-theoretical approach allows for a unified treatment of the diffusion and jump processes. Abstract results are illustrated by several examples and counter-examples.