Right Markov processes and systems of semilinear equations with measure data (Q259219)

Let \(E\) be a Radon metrizable topological space, \(F:\;E\times \mathbb{R}^{N}\to \mathbb{R}^{N}\), \(N\geq1\), be a measurable function and let \(\mu\) be a smooth measure on \(E\). The author proves the existence of probabilistic solutions to systems of the form \(-Au=F(x,u)+\mu\), where \(F\) is continuous with respect to \(u\) and satisfies the sign condition \(\langle F(x,y),y\rangle\leq G(x)| y|\), \(x\in E\), \(y\in \mathbb{R}^{N}\) for some appropriately integrable positive function \(G\); \(A\) is a generator of a Markov semigroup determined by a right Markov process whose resolvent is order compact on \(L^1\). The author studies some stability properties of the compactness property with respect to transformation of the underlying process. Let \({\mathcal B}(E)\) be the set of all numerical Borel measurable functions on \(E\). The author proves that, for every \(B\in {\mathcal B}(E)\), if \((X,{\mathcal P},m)\) has the compactness property, then \((X^{B},{\mathcal P}(B),m)\) has the compactness property, where \(X^{B}\) denotes the part of \(X\) on \(B\) and \({\mathcal P}(B)=\{u \in{\mathcal P}: u(x)=0, x\in E\setminus B\}\). Conditions on a sequence \(\{u_{n}\}\) of functions on \(E\), which together with the compactness property imply that \(\{u_{n}\}\) is relatively compact in the topology of \(m\) a.e. convergence, are found. The definition of a probabilistic solution is presented. The author first studies the probabilistic solution to the considered system in the case \(X\) is associated with a semi-Dirichlet form and \((X,{\mathcal B}_1, m)\) has the compactness property, and then the case of general right Markov processes is considered. Some examples of operators and processes to which the obtained results are applicable are presented.