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

Adam Bobrowski, and Radosław Bogucki. "Semigroups generated by convex combinations of several Feller generators in models of mathematical biology." Studia Mathematica 189.3 (2008): 287-300. <http://eudml.org/doc/286612>.

@article{AdamBobrowski2008,
abstract = {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.},
author = {Adam Bobrowski, Radosław Bogucki},
journal = {Studia Mathematica},
keywords = {contraction semigroup; generator; convex combination; approximation formulae; Trotter–Kato theorem; degenerate convergence; Feller process; weak convergence of processes; gene expression; gene regulation; fish dynamics},
language = {eng},
number = {3},
pages = {287-300},
title = {Semigroups generated by convex combinations of several Feller generators in models of mathematical biology},
url = {http://eudml.org/doc/286612},
volume = {189},
year = {2008},
}

TY - JOUR
AU - Adam Bobrowski
AU - Radosław Bogucki
TI - Semigroups generated by convex combinations of several Feller generators in models of mathematical biology
JO - Studia Mathematica
PY - 2008
VL - 189
IS - 3
SP - 287
EP - 300
AB - 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.
LA - eng
KW - contraction semigroup; generator; convex combination; approximation formulae; Trotter–Kato theorem; degenerate convergence; Feller process; weak convergence of processes; gene expression; gene regulation; fish dynamics
UR - http://eudml.org/doc/286612
ER -