Stochastic sequences with a regenerative structure that may depend both on the future and on the past

TL;DR

This paper develops a unified theory for stochastic processes with regenerative structures that depend on both past and future, providing new insights and applications to particle systems and ergodic theory.

Contribution

It introduces a general framework for conditioning on future and past in stochastic processes, with novel applications to contact processes and infinite-bin models.

Findings

New results for contact processes and infinite-bin models

Unified approach to conditioning on future and past

Connections established with Harris ergodicity

Abstract

Many regenerative arguments in stochastic processes use random times which are akin to stopping times, but which are determined by the future as well as the past behaviour of the process of interest. Such arguments based on "conditioning on the future" are usually developed in an ad-hoc way in the context of the application under consideration, thereby obscuring underlying structure. In this paper we give a simple, unified and more general treatment of such conditioning theory. We further give a number of novel applications to various particle system models, in particular to various flavours of contact processes and to infinite-bin models. We give a number of new results for existing and new models. We further make connections with the theory of Harris ergodicity.