Some Problems in Stochastic Flows and Couplings of Diffusions

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The principal investigator will study diffusions on manifolds and stochastic flows. The work on diffusions is aimed at various aspects of coupling and their applications. One goal is to develop a method for coupling a diffusion which arises from a Dirichlet form. A potential application of this work is a probabilistic proof of the Moser Harnack inequality. Another goal is to study the efficiency of the commonly used mirror coupling on manifolds. A goal of this work is to gain potential new insights of the influence of geometry on the bottom of the spectrum. A third problem is to determine which diffusions with the normalized Riemannian volume as equilibrium distribution converge to equilibrium the fastest. The principal problems in flows focus on rates of dispersion of bodies under stochastic flows. This research concentrates on the diffusion of particles on geometric structures called manifolds. It arises from the theory of Brownian motion and exploits well known connections between diffusions and the spread of heat in bodies (manifolds). Part of the research deals with rates of change of temperature distributions on manifolds while another deals with the rate of convergence to a steady state temperature distribution. These can be studied by diffusion theory and the influence of the geometry of the manifold on these quantities can be determined. The work on dispersion of stochastic flows is essentially the determination of how fast an oil slick will spread if all the oil particles are undergoing diffusive or Brownian motion. Applications of this work can be made to pollution control problems.