  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.
