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

Adam Bobrowski¹ ; Radosław Bogucki²

¹ Institute of Mathematics Polish Academy of Sciences Katowice branch Bankowa 14 40-007 Katowice, Poland on leave from Department of Mathematics Faculty of Electrical Engineering and Computer Science Lublin University of Technology Nadbystrzycka 38A 20-618 Lublin, Poland
² Actuarial and Insurance Solutions Deloitte Advisory Ltd. Pi/ekna 18 00-549 Warszawa, Poland

Studia Mathematica, Tome 189 (2008) no. 3, pp. 287-300

Résumé

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.

Détail

DOI : 10.4064/sm189-3-6

Keywords: mathcal locally compact hausdorff space ldots generators feller semigroups mathcal related feller processes geq alpha ldots non negative continuous functions mathcal sum alpha assume closure sum alpha defined bigcap mathcal generates feller semigroup geq mathcal natural interpretation related feller process geq evolves according following heuristic rules conditional being point mathcal probability alpha process behaves ldots provide approximation geq via sequence semigroups acting cartesian product copies cal supports interpretation generalizing main theorem bobrowski nbsp evolution equations where treated result motivated examples mathematical biology involving models gene expression gene regulation fish dynamics