We apply proof-theoretic techniques of ‘proof mining’ to obtain an effective uniform rate of metastability in the sense of Tao for Baillon’s famous nonlinear ergodic theorem in Hilbert space. In fact, we analyze a proof due to Brézis and Browder of Baillon’s theorem relative to the use of weak sequential compactness. Using previous results due to the author we show the existence of a bar recurs...

With my appreciation, I wish to thank my advisor Prof. Hillel Furstenberg for his guidance, support and encouragement during these unforgettable years of learning and research Abstract We associate ergodic properties to some subsets of the natural numbers. For any given family of subsets of the natural numbers one may study the question of occurrence of certain " algebraic patterns " in every s...

In this paper, we introduce a new class of processes which are diffusions with jumps driven by multivariate nonlinear Hawkes process. Our goal is to study their long-time behavior. the case exponential memory kernels for underlying process establish conditions positive Harris recurrence couple $(X,Y )$, where $X$ denotes diffusion and $Y$ piecewise deterministic Markov (PDMP) defining stochasti...

This paper is another case study in the program of logically analyzing proofs to extract new (typically effective) information (‘proof mining’). We extract explicit uniform rates of metastability (in the sense of T. Tao) from two ineffective proofs of a classical theorem of F.E. Browder on the convergence of approximants to fixed points of nonexpansive mappings as well as from a proof of a theo...

Ergodic systems, being indecomposable are important part of the study dynamical systems but if a system is not ergodic, it natural to ask following question: Is possible split into ergodic in such way that former reduces latter ones? Also, will be interesting see ones inherit some properties one. This document answers this question for measurable maps defined on complete separable metric spaces...

As a cornerstone of functional analysis, Hahn–Banach theorem constitutes an indispensable tool of modern analysis where its impact extends beyond the frontiers of linear functional analysis into several other domains of mathematics, including complex analysis, partial differential equations and ergodic theory besides many more. The paper is an attempt to draw attention to certain applications o...

We investigate the stability, in terms of V -uniform ergodicity or transience, of cyclic threshold autoregressive time series models. These models cycle through one of a number of collections of subregions of the state space when the process is large. Our results can be applied in cases where the model has multiple cycles and/or affine thresholds. The bounds on the parameter space are sharper t...

It has been established under very general conditions that the ergodic properties of Markov processes are inherited by their conditional distributions given partial information. While the existing theory provides a rather complete picture of classical filtering models, many infinite-dimensional problems are outside its scope. Far from being a technical issue, the infinite-dimensional setting gi...