Nonequilibrium Markov processes conditioned on large deviations

TL;DR

This paper develops a method to represent Markov processes conditioned on rare, large deviation events using a driven process constructed via a generalized Doob's h-transform, establishing equivalence in the long-time limit.

Contribution

It introduces a generalized Doob's h-transform to construct driven processes for conditioned Markov processes, linking large deviations, importance sampling, and nonequilibrium statistical mechanics.

Findings

Constructed driven process via generalized Doob's h-transform.

Proved long-time equivalence between conditioned and driven processes.

Linked driven process to exponential tilting and importance sampling.

Abstract

We consider the problem of conditioning a Markov process on a rare event and of representing this conditioned process by a conditioning-free process, called the effective or driven process. The basic assumption is that the rare event used in the conditioning is a large deviation-type event, characterized by a convex rate function. Under this assumption, we construct the driven process via a generalization of Doob's $h$-transform, used in the context of bridge processes, and show that this process is equivalent to the conditioned process in the long-time limit. The notion of equivalence that we consider is based on the logarithmic equivalence of path measures and implies that the two processes have the same typical states. In constructing the driven process, we also prove equivalence with the so-called exponential tilting of the Markov process, which is used with importance sampling to simulate rare events, and which gives rise, from the point of view of statistical mechanics, to a nonequilibrium version of the canonical ensemble. Other links between our results and the topics of bridge processes, quasi-stationary distributions, stochastic control, and conditional limit theorems are mentioned.