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

Adam Bobrowski ; Radosław Bogucki

Studia Mathematica, Tome 187 (2008), p. 287-300 / Harvested from The Polish Digital Mathematics Library

Résumé

Let be a locally compact Hausdorff space. Let $A_{i}$, i = 0,1,...,N, be generators of Feller semigroups in C₀() with related Feller processes $X_{i} = {X_{i}(t), t ≥ 0}$ and let $α_{i}$, i = 0,...,N, be non-negative continuous functions on with $∑_{i=0}^{N} α_{i} = 1$. Assume that the closure A of $∑_{k=0}^{N} α_{k}A_{k}$ defined on $⋂_{i=0}^{N} (A_{i})$ generates a Feller semigroup T(t), t ≥ 0 in C₀(). 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 ∈ , 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() 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

Publié le : 2008-01-01

Zbl 1161.47026

EUDML-ID : urn:eudml:doc:286612