Stochastic Operators and Semigroups and Their Applications in Physics and Biology

Abstract

Stochastic operators are positive linear operators defined on the space of integrable functions preserving the set of densities. They appear in many fields of mathematics and applications and they are use to study ergodic properties of dynamical systems and describe the evolution of Markov chains. Stochastic semigroups are continuous semigroups of stochastic operators. They describe the behaviour of the distributions of Markov processes like diffusion processes, piecewise deterministic processes and hybrid stochastic processes. In this chapter we present many examples of stochastic operators and semigroups: The Frobenius–Perron operator, diffusion semigroups, flows semigroups with jumps and switching and semigroups related to hybrid systems. Then we present some results concerning their long-time behaviour: asymptotic stability, sweeping, completely mixing and convergence to self-similar solutions. The results concerning stochastic operators are applied to study ergodicity, mixing and exactness of dynamical systems and an integral operator appearing in the theory of cell cycle. The general results concerning stochastic semigroups are applied to diffusion processes, jump processes and biological models described by piecewise deterministic stochastic processes: birth-death processes, the evolution of the genome, gene expression and physiologically structured models.