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Streszczenie

Let $\mathcal{S}$ be a locally compact Hausdorff space. Let $A_i,$ $i=0,1,\ldots,N$, be generators of Feller semigroups in $C_0(\mathcal{S})$ with related Feller processes $X_i = \{X_i(t), t \geq 0\}$ and let $\alpha_i,$ $i =0,\ldots,N$, be non-negative continuous functions on $\mathcal{S}$ with $\sum_{i=0}^N\alpha_i= 1.$ Assume that the closure $A$ of $\sum_{k=0}^N \alpha_k A_k$ defined on $\bigcap_{i=0}^N \mathcal{D}(A_i)$ generates a Feller semigroup $\{T(t), t \geq 0\}$ in $C_0(\mathcal{S}).$ A natural interpretation of a related Feller process $X= \{X(t), t \geq 0\}$ is that it evolves according to the following heuristic rules: conditional on being at a point $p \in \mathcal{S},$ with probability $\alpha_i (p),$ the process behaves like $X_i,$ $i=0,1,\ldots,N.$ We provide an approximation of $\{T(t), t \geq 0\}$ via a sequence of semigroups acting in the Cartesian product of $N+1$ copies of $C_0({\cal S})$ that supports this interpretation, thus generalizing the main theorem of Bobrowski [J. Evolution Equations 7 (2007)] where the case $N=1$ is treated. The result is motivated by examples from mathematical biology involving models of gene expression, gene regulation and fish dynamics.