Brownian Motion and the Dirichlet Problem at Infinity on Two-dimensional Cartan-Hadamard Manifolds

After recalling the Dirichlet problem at infinity on a Cartan-Hadamard manifold, we describe what is known under various curvature assumptions and the difference between the two-dimensional and the higher-dimensional cases. We discuss the probabilistic formulation of the problem in terms of the asymptotic behavior of the angular component of Brownian motion. We then introduce a new (and appealing) probabilistic approach that allows us to prove that the Dirichlet problem at infinity on a two-dimensional Cartan-Hadamard manifold is solvable under the curvature condition K ≤ (1 + ε)/(r2 logr) outside of a compact set, for some ε > 0, in polar coordinates around some pole. This condition on the curvature is sharp, and improves upon the previously known case of quadratic curvature decay. Finally, we briefly discuss the issues which arise in trying to extend this method to higher dimensions.