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Modification of balayage spaces by transitions with application to coupling of PDE’s. (English) Zbl 1094.31005
The author considers modifications of balayage spaces which correspond to killing and transitions (creation of mass combined with jumps) in the Markov process setting. Let $$(X, {\mathcal W})$$ be a balayage space associated with a family $$(H_U)_{U\in{\mathcal U}}$$ of regular harmonic kernels and $$K_X$$ be a potential kernel for $$(X,{\mathcal W})$$. Let $$T$$ be an admissible transition kernel on $$X$$ and $$k$$ be a Kato function on $$X$$ with respect to $$K_X$$. A modification of $$(X,{\mathcal W})$$ by $$T$$ and $$k$$ is $$(X,\widetilde{{\mathcal W}}^T)$$, where $\widetilde{{\mathcal W}}^T:=\{v;v\: X\rightarrow[0,\infty], \text{ l.s.c.}, \widetilde{H}^T_U v\leq v \text{ for every } U\in{\mathcal U}^T\}.$ Here $${\mathcal U}^T$$ is the set of all $$U \in {\mathcal U}$$ such that $$T$$ is a transition from $$U$$ to the complement of $$U$$ and $\widetilde{H}^T_U := (I + K_UM_k)^{-1}(H_U + K_UT).$ The author gives some sufficient conditions for $$(X, \widetilde{{\mathcal W}}^T)$$ to be a balayage space. For example, if $$1 \in {\mathcal W}$$, $$k > T1$$ and for every $$U \in {\mathcal U}$$, $$K_U 1$$ is strictly positive on $$U$$, then $$(X, \widetilde{\mathcal W}^T)$$ is a balayage space. The results can be applied to study PDEs. The general theory of balayage space gives an immediate solution to the Dirichlet problem for coupled PDEs.

##### MSC:
 31D05 Axiomatic potential theory 60J25 Continuous-time Markov processes on general state spaces 35J55 Systems of elliptic equations, boundary value problems (MSC2000)
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