Nonequilibrium Markov Processes Conditioned on Large Deviations

By R. Chetrite, Hugo Touchette. 2014

342 citations recorded by the source corpus.

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, often used with importance sampling to simulate rare events and giving 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.

Subfields: Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e5cc65cf7aa6f2d92fc4059ea5bfea5a55ae95ef

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In Statistical and Nonlinear Physics

In Statistics and Probability

Hugo Touchette

Citations in the corpus

Introduction to dynamical large deviations of Markov processes (2017, background): “Sources: [27, 28].”
Current fluctuations and transport efficiency for general Langevin systems (2017, background): “To this end, we define a generalized time-integrated current R(t) via its time derivative [26] Ṙ(t) = Z(x(t), t) ◦ ẋ(t), (3)”