Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Computer Science


Theoretical Statistics and Mathematics Unit (TSMU-Bangalore)


Ramasubramaniam, S. (TSMU-Bangalore; ISI)

Abstract (Summary of the Work)

An attempt to obtain conditions for certain stability properties of reflecting diffusions in unbounded domains with boundary has been made in this thesis. For diffusions in R', such stability properties like recurrence, transience and positive recurrence have been studied extensively; see Bhattacharya (1978), Kliemann (1987), Pinsky (1987). One might see Pinsky (1995) for an up-to-date review of kuown methods and results in this all case. (For corresponding recurrence classification results on Markov chains using martin- gale ideas based on stoxchastic analogues of Lyapunov functions, see Meyn and Tweedie (1993a), (1993b) and the references given therein). The main concern in this study is to establish related results in the case of unbounded domains like the half space, the orthant and the quadrant.Natural definitions of recurrence, transience, and positive recurrence are used throughout and they will be stated precisely in Chapter 2. In general, the diffusion is said to be recurrent if it visits every neighborhood of the starting point infinitely often. It is said to be transient if starting from any point it wanders off to infinity. A diffusion is positive rerurrent if the hitting time of any bounded open set has a finite expretation. In the case of recurrent (positive recurrent) diffusion, the existence of a unique o-finite (finite) invariant measure can be shown.In the case of a sınooth bounded domain, the reflecting diffusion, being a Feller continuous strong Markov process on a compact space, has an invariant probability measure and hence is positive recurrent. Therefore, the problem of interest is in unbounded domains.In Chapter 2, reflecting diffusions in the half space are considered, where the dispersion and drift are Lipschitz continuous funetions and the rellertion lield is C"-Nmoot.h (see, however, Remark 2.1.10). The existence and uniquencss of such diffusions is well known. A dichotomy between recurrence and transience of such reflecting diffusious is proved; (a priori such a dichotomy is not obvious). We give proofs only when it differs from the case of diffusions (in R) without boundary conditions. (soe case(ii) in the proof of Lemma 2.1.2(a) and the proof of (e) (d) in Proposition 2.1.3). The main difference is the following: It is not clear if au analogue of Lemma 2.3(b) of Bhattacharya (1978) holds in the case of reflecting diffusions. (Of course, maximum principles under stronger differentiability conditions are available as in Protter and Weinberger (1967)). Further, functional characterisations of recurrence and transience are obtained which in turn lead to verifiable sufficient criteria for recurrence/transience in terms of appropriate Lyapunov functions. Using these criteria a real variable proof of certain interesting results of Rogers (1991) concerning reflecting Browniau motion in the half plane are given. Also, it is shown that the hitting time of any open set. has a finite expctation if there is one positive recurrent point; in the course of the proof an analogue of an estimate due to Dupuis-Willians (1994) is obtained. The problem of transience down a side in the case of reflecting diffusious in the half plane is also dealt with.


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