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In the paper we prove the existence of probabilistic solutions to systems ofthe form $-Au=F(x,u)+\mu$, where $F$ satisfies a generalized sign condition and$\mu$ is a smooth measure. As for $A$ we assume that it is a generator of aMarkov semigroup determined by a right Markov process whose resolvent is ordercompact on $L^1$. This class includes local and nonlocal operatorscorresponding to Dirichlet forms as well as some operators which are not in thevariational form. To study the problem we introduce new concept of compactnessproperty relating the underlying Markov process to almost everywhereconvergence. We prove some useful properties of the compactness property andprovide its characterization in terms of Meyer's property (L) of Markovprocesses and in terms of order compactness of the associated resolvent.

Right Markov processes and systems of semilinear equations with measure data