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Let E be a noncompact locally compact second countable Hausdorff space. We consider the question when, given a family of finite nonzero measures on E that behave like harmonic measures associated with all relatively compact open sets in E (i.e. that satisfy a certain consistency condition), one can construct a Markov process on E and a multiplicative functional with values in [0, ∞) such that the hitting distributions of the process “inflated” by the multiplicative functional yield the given harmonic measures. We achieve this construction under weak continuity and local transience conditions on these measures that are natural in the theory of Markov processes, and a mild growth restriction on them. In particular, if the space E equipped with the measures satisfies the conditions of a harmonic space, such a Markov process and associated multiplicative functional exist. The result extends in a new direction the work of many authors, in probability and in axiomatic potential theory, on constructing Markov processes from given hitting distributions (i.e. from harmonic measures that have total mass no more than 1).

Construction of Markov processes and associated multiplicative functionals from given harmonic measures