Markov Processes Generated by Random Iterates of Monotone Maps: Theory and Applications

An Approach to the Existence of Unique Invariant Probabilities for Markov Processes

A notion of localized splitting is introduced as a further extension of the splitting notions for iterated monotone maps introduced earlier by Dubins and Freedman (1966) and more generally by Bhattacharya and Majumdar (1999). We will see that under quite general conditions, localized splitting theory is a natural extension of the Doeblin (1937) minorization theory, Harris (1956) recurrence theo...

Random Iterates of Monotone Maps

Dynamical Behavior of a Stochastic Forward-Backward Algorithm Using Random Monotone Operators

The purpose of this paper is to study the dynamical behavior of the sequence produced by a forward-backward algorithm, involving two random maximal monotone operators and a sequence of decreasing step sizes. Defining a mean monotone operator as an Aumann integral, and assuming that the sum of the two mean operators is maximal (sufficient maximality conditions are provided), it is shown that wit...

Continuous dependence on coefficients for stochastic evolution equations with multiplicative Levy Noise and monotone nonlinearity

Semilinear stochastic evolution equations with multiplicative L'evy noise are considered‎. ‎The drift term is assumed to be monotone nonlinear and with linear growth‎. ‎Unlike other similar works‎, ‎we do not impose coercivity conditions on coefficients‎. ‎We establish the continuous dependence of the mild solution with respect to initial conditions and also on coefficients. ‎As corollaries of ...

The Quasi-Stationary Distribution for Small Random Perturbations of Certain One-Dimensional Maps

We analyze the quasi-stationary distributions of the family of Markov chains {Xε n }, ε > 0, obtained from small non-local random perturbations of iterates of a map f : I → I on a compact interval. The class of maps considered is slightly more general than the class of one-dimensional Axiom A maps. Under certain conditions on the dynamics, we show that as ε → 0 the limit quasi-stationary distri...