engleski

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.

standard process ; harmonic functions ; Harnack inequality ; trace process

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