The Augmented Jump Chain -- a sparse representation of time-dependent Markov jump processes

TL;DR

This paper introduces the augmented jump chain, a sparse, space-time representation of non-autonomous Markov jump processes, enabling efficient analysis of time-dependent dynamics in high-dimensional systems.

Contribution

It develops a novel augmented jump chain framework that extends Markov jump processes to non-autonomous systems using space-time embedding, with theoretical foundations and practical discretization methods.

Findings

The augmented jump chain inherits sparsity from the original process.

It facilitates analysis of non-autonomous dynamics in high dimensions.

Proof-of-concept discretization demonstrates practical applicability.

Abstract

Modern methods of simulating molecular systems are based on the mathematical theory of Markov operators with a focus on autonomous equilibrated systems. However, non-autonomous physical systems or non-autonomous simulation processes are becoming more and more important. We present a representation of non-autonomous Markov jump processes as autonomous Markov chains on space-time. Augmenting the spatial information of the embedded Markov chain by the temporal information of the associated jump times, we derive the so-called augmented jump chain. The augmented jump chain inherits the sparseness of the infinitesimal generator of the original process and therefore provides a useful tool for studying time-dependent dynamics even in high dimensions. We furthermore discuss possible generalizations and applications to the computation of committor functions and coherent sets in the non-autonomous setting. After deriving the theoretical foundations we illustrate the concepts with a proof-of-concept Galerkin discretization of the transfer operator of the augmented jump chain applied to simple examples.