|Ph.D Student||Vadlamani Sreekar|
|Subject||On the Diffusion of Shape|
|Department||Department of Industrial Engineering and Management||Supervisor||Professor Leonid Mytnik|
|Full Thesis text|
The main objective of this thesis is to study the global geometric properties of a manifold embedded in Euclidean space, as it evolves under a stochastic flow of diffeomorphisms. The processes driving the stochastic flows are chosen to be Gaussian processes with stationary increments (in time). The most common class of Gaussian processes with stationary increments is the family of fractional Brownian motions with Hurst parameter H taking values in (0,1). This family encompasses a wide variety of processes with applications in the fields of oceanography, finance and telecommunications, to name a few. The fact that these processes possess stationary increments implies that the corresponding noise process is a stationary process, and so one can hope to obtain ergodic estimates.
In Part I of the dissertation, we study the evolution of a co-dimension one manifold embedded in Euclidean space, under an
isotropic and volume preserving Brownian flow. In particular we obtain expressions describing the expected rate of growth of the
Lipschitz-Killing curvatures, or intrinsic volumes, of the manifold evolving under the flow. These results shed new light on the some of the intriguing growth properties of flows from a global perspective, rather than the local perspective, on which there is much larger literature.
In Part II, we deviate from the setting of standard Brownian flows, whose analysis was primarily based on the Markovian character of the flow, and move to stochastic flows driven by fractional Brownian motion with Hurst parameter H taking values in (1/2, 1). Adopting a pathwise approach, we obtain estimates for the growth of the Hausdorff measure of an m-dimensional manifold embedded in n-dimensional Euclidean space.