Variational and optimal control representations of conditioned and driven processes

TL;DR

This paper presents new variational and optimal control representations of driven processes, which describe conditioned Markov processes, unifying previous approaches and enabling new analytical and numerical methods for large deviation functions.

Contribution

It introduces two novel representations of driven processes: as variational principles involving large deviations and as optimal control problems, extending existing theories.

Findings

Driven processes can be represented via variational principles.

Driven processes can be formulated as optimal stochastic control problems.

These representations unify and extend previous theoretical frameworks.

Abstract

We have shown recently that a Markov process conditioned on rare events involving time-integrated random variables can be described in the long-time limit by an effective Markov process, called the driven process, which is given mathematically by a generalization of Doob's $h$-transform. We show here that this driven process can be represented in two other ways: first, as a process satisfying various variational principles involving large deviation functions and relative entropies and, second, as an optimal stochastic control process minimizing a cost function also related to large deviation functions. These interpretations of the driven process generalize and unify many previous results on maximum entropy approaches to nonequilibrium systems, spectral characterizations of positive operators, and control approaches to large deviation theory. They also lead, as briefly discussed, to new methods for analytically or numerically approximating large deviation functions.