Mathematical Sciences: Geometric Aspects of Random Motion

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9626428 Cranston ABSTRACT The principal investigator attacks several problems in the general area of Stochastic Analysis. One problem is to come up with a coupling for diffusions with degenerate generators given by sums of squares of vector which satisfy Hormander's hypoellipticity condition. A related problem is to give a coupling proof of Moser's Harnack inequality. The principal investigator also uses coupling techniques on Brownian motion on Gromov hyperbolic spaces to estimate eigenvalues and gradients of harmonic functions. The principal investigator also studies relativistic Brownian motion. Finally, the investigator is deriving more results on the behavior of processes with dependence on the past. The work in the current proposal falls into the general area of Stochastic Analysis. This means the analysis of randomly evolving phenomena. The portions of this proposal on coupling are related to the smoothness of temperature distributions on surfaces, their rate of convergence to a steady state and the dependence of these on the geometry of the surface. The problems on processes whose behavior depends on the past have potential applications to economics (prices e.g.) and to biology (chemotactic phenomena.) The work on relativistic Brownian motion can have potential applications to cosmology, especially in deciding whether it is possible to observe black holes.