Graduate Student Seminar: Stochastic Dynamics through Hierarchically Embedded Markov Chains - Extending the Rare Mutation Limit, Speaker: Vítor V. Vasconcelos

Title:  Stochastic Dynamics through Hierarchically Embedded Markov Chains - Extending the Rare Mutation Limit

Abstract:   Studying dynamical phenomena in finite populations often involves Markov processes of significant mathematical and/or computational complexity, which rapidly becomes prohibitive with increasing population size or an increasing number of individual configuration states. Here, we develop a framework that allows us to define a hierarchy of approximations to the stationary distribution of general systems that can be described as discrete Markov processes with time invariant transition probabilities and (possibly) a large number of states, in particular in the presence of quasi-invariant manifolds as a control parameter tends to zero. This results in an efficient method for studying social and biological communities in the presence of stochastic effects — such as mutations, in evolutionary dynamics, and random exploration of choices, in social systems — including situations where the dynamics encompasses the existence of stable polymorphic configurations, thus overcoming the limitations of existing methods. The present formalism is shown to be general in scope, widely applicable, and of relevance to a variety of interdisciplinary problems.