Asymptotic decomposition of substochastic semigroups and applications

Theorems on asymptotic decomposition of substochastic semigroups are given. General results are applied to show asymptotic properties of substochastic semigroups generated by dynamical systems with random switching.

On the Honesty in Nonlocal and Discrete Fragmentation Dynamics in Size and Random Position

A discrete initial-value problem describing multiple fragmentation processes, where the fragmentation rate is size and position dependent and where new particles are spatially randomly distributed according to some probabilistic law, is investigated by means of parameter-dependent operators together with the theory of substochastic semigroups with a parameter. The existence of semigroups is established for each parameter and “glued” together so as to obtain a semigroup to the full space. Under certain conditions on each fragmentation rate, we used Kato’s Theorem in to show the existence of the generator and we provide sufficient conditions for honesty.