The first part of this research will deal with Brownian motion on negatively curved manifolds. The investigator intends to find the Martin boundary for strongly negatively curved manifolds, and identify the bounded solutions of the heat equation. Also, conditions will be found under which harmonic measure on such manifolds is absolutely continuous with respect to angular measure, in polar coordinates. The second problem will deal with a nonlinear stochastic version of the heat equation. The investigator will study the rate at which the solution decays to zero. This will be a first step toward studying the qualitative behavior of such equations. The third problem will deal with the law of iterated logarithm for analytic functions, recently obtained by Markov. A version of Chung's law of the iterated logarithm will be sought for analytic functions. From this, packing dimension of the support of harmonic measure might be found in the same way as Markov found the Hausdorff dimension of the support.
