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Semigroups generated by convex combinations of several Feller generators in models of mathematical biology

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Let 𝓢 be a locally compact Hausdorff space. Let $A_{i}$, i = 0,1,...,N, be generators of Feller semigroups in C₀(𝓢) with related Feller processes $X_{i} = {X_{i}(t), t ≥ 0}$ and let $α_{i}$, i = 0,...,N, be non-negative continuous functions on 𝓢 with $∑_{i=0}^{N} α_{i} = 1$. Assume that the closure A of $∑_{k=0}^{N} α_{k}A_{k}$ defined on $⋂_{i=0}^{N} 𝓓(A_{i})$ generates a Feller semigroup {T(t), t ≥ 0} in C₀(𝓢). A natural interpretation of a related Feller process X = {X(t), t ≥ 0} is that it evolves according to the following heuristic rules: conditional on being at a point p ∈ 𝓢, with probability $α_{i}(p)$, the process behaves like $X_{i}$, i = 0,1,...,N. We provide an approximation of {T(t), t ≥ 0} via a sequence of semigroups acting in the Cartesian product of N + 1 copies of C₀(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.
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  • Institute of Mathematics, Polish Academy of Sciences, Katowice branch, Bankowa 14, 40-007 Katowice, Poland
  • Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin, Poland
  • Actuarial and Insurance Solutions, Deloitte Advisory Ltd., Piękna 18, 00-549 Warszawa, Poland
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm189-3-6
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