Nonequilibrium Markov processes conditioned on large deviations
The authors give detailed considerations about the problem of conditioning a Markov process on a rare event. This conditioned process is represented by a conditioning free process that is named the effective or driven process. The whole approach is based on the assumption that the rare event (used in the conditioning) is a large deviation-type event with convex rate function. The driven process is constructed via a generalization of Doob's \(h\)-transform and the authors show that it is equivalent to the conditioned process in the long-time limit. The mentioned equivalence is based on the logarithmic equivalence of path measures. The construction of the driven process allows to prove the equivalence with the so-called exponential tilting of the Markov process. It is used with importance sampling to simulate rare events and giving rise to a non-equilibrium version of the canonical ensemble. The whole paper is divided into six sections and five appendices. After the introduction and a justification of the proposed work, the assumed notations and definitions are presented in Section 2. Then Section 3 discusses the properties of a non-conservative process associated with the canonical path measure study. The authors' approach is based on Doob's transform, thus in Section 4 they present a generalization of this transform that is used in Section 5 to define the driven process. The last section shows possible applications of the proposed approach based on the study of Brownian motion, the Ornstein-Uhlenbeck process, and the problem of quasi-stationary distributions. In the appendices, all necessary details needed in the paper are presented.
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