Narrow Results

We consider Vlasov-type scaling for the Glauber dynamics in continuum with a positive integrable potential, and construct rescaled and limiting evolutions of correlation functions. Convergence to the limiting evolution for the positive density system in infinite volume is shown. Chaos preservation property of this evolution gives a possibility to derive a nonlinear Vlasov-type equation for the particle density of the limiting system.

We study a Markov birth-and-death process on a space of locally finite configurations, which describes an ecological model with a density-dependent fecundity regulation mechanism. We establish existence and uniqueness of this process and analyze its properties. In particular, we show global time-space boundedness of the population density and, using a constructed Foster–Lyapunov-type function, we study return times to certain level sets of tempered configurations. We also find sufficient conditions that the degenerate invariant distribution is unique for the considered process.

Evolutionary dynamics studies changes in populations of species, which occur due to various processes such as replication and mutation. Here we consider this dynamics as an example of Markov evolution on a simplex of probability measures describing the populations, and then define optimality of this evolution with respect to constraints on information distance between these measures. We show how this convex programming problem is related to a variational problem of optimizing Markov transition kernel subject to a constraint on Shannon's mutual information. This relation is represented by the Pythagorean theorem in information geometry considered on the simplex of joint probability measures. We discuss the application of this variational approach to optimization of a stochastic search in metric spaces, and in particular to optimization of mutation rate parameter during the search for optimal DNA sequences in evolutionary systems.