Non-trivial bounded harmonic functions on Cartan-Hadamard manifolds of unbounded curvature

In this thesis we study the existence of non-trivial bounded harmonic functions on certain Cartan-Hadamard manifolds $M$ for which the Dirichlet Problem at infinity is not solvable. Hereby we use the fact that there is a one-to-one correspondence between the space of all bounded harmonic functions on $M$ and the set of all bounded functions, which are measurable with respect to the shift invariant $\sigma$-field, up to equivalence. Ancona (1994, using probabilistic methods) and Borbely (1998 with analytic methods) constructed examples of Cartan-Hadamard manifolds for which the Dirichlet problem at infinity is not solvable. We show that almost surely Brownian motion on the manifold of Borbely exits from the manifold at a single point at the sphere at infinity, i.e. from the probabilistic point of view the examples of Ancona and Borbely are essentially the same. Moreover we show that on both manifolds there are non-trivial exit sets for the Brownian motion,i.e. non-trivial shift invariant events and hence there exist non-trivial bounded harmonic functions. In the case of the manifold of Borbely the asymptotic behaviour of the Brownian motion even yields two "independent" non-trivial shift-invariant random-variables hence the Martin boundary has to be at least of dimension two. We give a geometric interpretation how this behaviour of the Brownian motion can be visualized via a change of coordinates. Last we use the manifold of Ancona to construct further examples of Cartan- Hadamard manifolds with unbounded curvature where the asymptotic behaviour of the Brownian motion can be "predetermined" whereas there exist non-trivial bounded harmonic functions.