Diffusion processes are often used to model the evolutions of various random phenomena in the natural and social sciences. Hitherto most of the research on diffusions concern nondegenerate diffusions. Their asymptotics have been extensively studied and are well understood. But the same is not true for the degenerate case. Recently, in Basak (1991) and Basak and Bhattacharya (1992), some qualitative theory have been developed to study the stability and functional limit theorems for degenerate diffusions....[ Read more ]

Diffusion processes are often used to model the evolutions of various random phenomena in the natural and social sciences. Hitherto most of the research on diffusions concern nondegenerate diffusions. Their asymptotics have been extensively studied and are well understood. But the same is not true for the degenerate case. Recently, in Basak (1991) and Basak and Bhattacharya (1992), some qualitative theory have been developed to study the stability and functional limit theorems for degenerate diffusions.

In this thesis we study the stability, functional limit theorems and ergodic control problem for a class of random degenerate diffusions. This is a typical hybrid system which arises in numerous applications such as fault tolerant control systems, flexible manufacturing systems and in many others. Our work extends the results of Basak and Bhattacharya. Under certain Liapunov type conditions we prove the existence of a unique invariant probability and stability in distribution of the flow. The functional limit theorem and the functional law of iterated logarithm have been derived for a broad class of functions belonging to the range of the infinitesimal generator of the random diffusion. We also consider a linear system with Markovian switching which is perturbed by Gaussian type noise. We show that under certain conditions the noise added system remains stable when (i) the linear system is mean square stable; (ii) the linear system is not mean square stable. Finally, we establish the existence of an optimal control for the ergodic control problem of random degenerate diffusions. We discuss the corresponding HJB equation and establish the existence of a unique viscosity solution in a certain class. A characterization of the optimal control in terms of the unique viscosity solution is obtained.