Brown'sche Bewegung auf Mannigfaltigkeiten / Brownian Motion on Manifolds

This lecture course deals with the basics of Brownian motion on a Riemannian manifold. After establishing the required abstract functional analytic facts (like the spectral theorem for unbounded operators), we define the Laplace-Beltrami operator on the Hilbert space of square integrable functions with respect to the volume measure on a Riemannian manifold, as well as the heat semigroup of this operator. The integral kernel of the latter semigroup (the so called heat kernel) then serves as the transition density of a diffusion: the celebrated Brownian motion. Finally, analytic applications of Brownian motion will be presented.