## Doctoral Theses

### Contributions to Random Interactions and Dynamical Systems.

2-22-2000

2-22-2001

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Computer Science

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Rao, B. V. (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

In recent years random iterations of maps on Polish spaces has gained prominence. They are. nice examples of Markov processes whose invariant measures can be used in Computer imaging (see Berger ( 1). They also arise as random perturabations of deterministic dynamical systems.Let S be a Polish space with its Borel a-field. Let r be a collection of Borel maps from S to S. Let P be a probability on r. Then starting with a point z in S, we choose a map yn er according to the law P and move to ya(x). Then we choose 2 â‚¬ r again according to the law P and move to 72 Y1 (x) and so on. This gives rise to a Markov process with state space S. One is interested in the existence and uniqueness of invariant measures for the process. Also when there is a unique invariant measure, one is interested in the nature of invariant measure like its support, whether it is absolutely continuous with respect to another given probability etc. Eventhough there were earlier works, L.E. Dubins and D.A. Freedman [12] for the first time made a systematic study when the Polish space S is real line. A part of their work was generalized to higher dimensions by R.N. Bhattacharya and his co-authors. The problems dealt with in this thesis are either directly or indirectly connected with random iterations. The thesis has four chapters. Each chapter starts with a summary of its own. We briefly describe the main contents below.In chapter I, we discuss the problem of completeness of a metric â€“ introduced by R.N. Bhattacharya and O. Lee [ 3] - on the space of probabilities on R*. This metric was introduced by them in generalizing the works of Du- bins and Freedman (12] regarding existence of invariant measures for Markov processes generated by random iterations of monotone maps. They obtained positive results bypassing the problem of completeness of the metric. They suggested that if the metric could be proved to be complete, then a fixed point theorem will make the arguments simpler. We carry out this programme. To generalize these results from R* to appropriate subsets S of R*, it is necessary to know for which subsets S of R*, the class of probabilities on S, say, P(S) is complete under the metric. However, we do not know the full answer to this question.In chapters II and II, we study a problem whose origins go back to the works of M. Rosenblatt (29). Given a probability u on Sa, the space of stochastic matrices of order d - which is a semigroup under multiplication - find conditions for the convolution sequence to converge. Several conditions in the general context of compact groups and semigroups were already available in Rosenblatt (29]. See A. Mukherjea and G. Hognas (17] for a thorough and update treatment. The question however is to find some simply verifiable conditions on u so that p nconverges. When d = = 2, this was treated by A. Mukherjea [22]. His theorem reads as follows. If Âµ is probability on the space of 2 x 2 stochastic matrices, then & converges if 0 1 1 0 and oly if u is not the point mass at the matrix

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