Adam Bobrowski (IM PAN) - On limitations and insufficiency of the Trotter-Kato theorem, with applications to a model of stochastic gene expression (2 lectures)

Abstract: Motivation for the talk comes from a recent model of stochastic gene expression introduced by Lipniacki et al. (J. Theor. Biol. 238 (2006), 348-367.) The model involves a family of Feller processes, solutions to systems of stochastic differential equations driven by Markov chains with state-dependent jump intensties, which naturally converge to a certain deterministic process. It turns out that convergence of related semigroups of operators cannot be proved by means of the classical Trotter--Kato theorem, and the difficulty lies in a somewhat unexpected place. Before we deal with this difficulty, to explain the source of the problem, we exhibit simple examples of convergence of equibounded semigroups that cannot be captured by means of the Trotter-Kato theorem. In this context we discuss the need for semigroup-theoretical tools that would supplement this theorem in dealing with convergence problems.