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

Nonequilibrium Markov processes conditioned on large deviations