On harmonic functions for trace processes

Let X be a standard Markov process with state space E and let F be a closed subset of E. A nonnegative function f on F is extended probabilistically to a function h_f on the whole space E. We show that the extension h_f is harmonic with respect to X provided that f is harmonic with respect to Y, the trace process on F of the process X. A consequence is that if the Harnack inequality holds for X, it also holds for the trace process Y. Several examples illustrating the usefulness of the result are given.