Havva Yoldas defenderá su tesis doctoral el viernes 11 de octubre

We study the long-time behaviour of solutions to some partial differential equations arising in modeling of several biological and physical phenomena. In this work, the type of the equations we consider is mainly nonlocal, in the sense that they involve integral operators. Moreover, the equations we consider describe the time evolution of either some populations structured by several traits like age, elapsed-time and size or the distribution of the dynamical states of a single particle, depending on time, space and velocity. In the latter case, they are called kinetic equations. The physiologically structured population models and the space inhomogeneous linear kinetic equations we deal with in this work are well-studied from various aspects in the already-existing literature. What differs from the past plentiful studies on the asymptotic behaviour of these equations is the techniques we use here. We consider a probabilistic approach which is first developed for studying ergodic properties of discrete-time Markov processes. The method is due to Doeblin and Harris; based on establishing a combination of a minorisation (irreducibility) and a geometric drift (Lyapunov) conditions for a Markovian process. This method gives a quantitative convergence speed and existence of a unique steady state even without having to calculate it explicitly. Application of Harris’s Theorem into the aforementioned partial differential equations to study the long-time behaviour of solutions is the core of this thesis.