The Dirichlet Problem of a Discontinuous Markov Process

Given a Markov process satisfying certain general type conditions, whose paths are not assumed to be continuous. Let D be an open subset of the state space E. Any bounded function defined on the complement of D extends to be a function on E such that it is harmonic in D and satisfies the Dirichlet boundary condition at any regular boundary point of D. The relation between harmonic functions and the characteristic operator of the given process is discussed.