Topics in multi-dimensional diffusion theory: Attainability, reflection, ergodicity and rankings

This thesis is a contribution to multi-dimensional diffusion theory. Attainability, ergodicity, and rankings of n-dimensional diffusions will be discussed in the intersection of the theories of elliptic partial differential equations and of the stochastic calculus. The idea of effective dimension for diffusions, originally explored in the theory of the exterior Dirichlet problem, gives a criterion for the attainability of an (n-2)-dimensional hyperplane.;This attainability can be rephrased as a triple-collision problem of n diffusive interacting particles on the real line. Another criterion for the attainability comes from the so-called skew-symmetry condition of Brownian motion with oblique reflection in the (n-1)-dimensional positive orthant. Non-attainability plays a crucial role not only in uniqueness of the diffusion in the sense of probability distribution but also in determining related one-dimensional local times of continuous semi-martingales.;These considerations have ramifications concerning the ergodic properties of ranked diffusion obtained from those of the (n-1)-dimensional Brownian motion with reflection. These topics will be united in a fresh manner with an application to the mathematical study of the Atlas model of equity market.