Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities (original) (raw)

Stochastic viability and invariance

Annali Scuola Normale di Pisa, 1990

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

2000

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, local- ized splitting theory is a natural extension of the Doeblin (1937) minoriza- tion theory, Harris (1956) recurrence

Existence and Uniqueness of Fixed Points for Markov Operators and Markov Processes

Proceedings of the London Mathematical Society, 1998

This paper concerns a Markov operator T on a space L I , and a Markov process P, which defines a Markov operator on a space M of finite signed measures. For T, the paper presents necessary and suficient conditions for: (a) the existence of invariant probability densities (IPDs) (b) existence of strictly positive IPDs, and (c) existence and uniqueness of IPDs. Similar results on invariant probability measures for P are presented. The basic approach is to pose a fixed-point problem as the problem of solving a certain linear equation in a suitable Banach space, and then obtain necessary and sufficient conditions for this equation to have a solution. A M S subject classification: 60505, 47B65, 47N30. K e y word a n d phrases. Markov operators, Markov processes, fixed points, invariant probability measures, invariant probability densities, linear equations in Banach space.

Capacity and energy for multiparameter Markov processes

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On the discrepancy of Markov-normal sequences

Journal de Theorie des Nombres de Bordeaux, 1996

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On a class of measures connected with past – dependent probability evolutions

2014

A class of probability measures on the space of the trajectories of a dynamical system with discrete time is introduced. The concepts of “homogeneity of a measures” and of “stationarity of a measure” are discussed for the elements of this class. And some elementary examples of such measures are studied in connection with the problem of describing those evolutions of probability which depend on the whole past distributions of the system. 1 Statement of the problem In the study of those systems whose evolutions is ruled by probabilistic laws, the probability measures on the “space of trajectories” present particular importance. For instance, Bernouilli’s measure, can be considered as a probability measure connected with a completely casual evolution. Markov’s measures, as W. Feller repeatedly stressed (cfr. [7]; pg. 420), may be considered as associated to the most direct probabilistic generalization of a deterministic evolution described by ordinary differential equations (i.e., a dy...

Self-similar vector measures of Markov-type operators

arXiv: Classical Analysis and ODEs, 2017

We consider iterated function systems (finite or countable), together with linear and continuous operators on Hilbert spaces, which enable us to construct Markov-type operators. Under suitable conditions, these Markov-type operators have fixed points, which are self-similar (invariant) vector measures, thus generalizing the classic Hutchinson self-similar measures. Several models with concrete computations are introduced.