Random Dynamical Systems and Selected Works of Rabi Bhattacharya

Abstract

The topic of random dynamical systems is extremely broad. However the focus of Rabi Bhattacharya’s work in this area is largely from the perspective of discrete parameter Markov processes on a general state space S, equipped with a suitable sigmafield $\mathcal{S}$ of measurable subsets. Such Markov processes are either prescribed as evolutions defined by i.i.d. iterated random maps from S to S, or by such a representation theorem that holds for any discrete parameter Markov processes having stationary transition probabilities on a Borel subset S of a Polish space, with Borel sigmafield $\mathcal{S}$. A theme of much of Rabi’s work is that of existence and uniqueness of invariant probabilities under conditions in which the Markov process may not be irreducible. These and corresponding problems concerning rates of convergence and various asymptotic limit theorems are representative of the research addressed here. Applications, particularly to geosciences and economics, are also a main theme of Rabi’s body of work in this area; however, these will be covered in separate essays and not treated here. The co-authored texts [6, 11] include a variety of such applications.