The most well-studied example of a stochastic process is Brownian motion. The deep connection between the Laplacian as being the generator of Brownian motion (up to a constant) naturally extends to the setting of (smooth Riemannian) manifolds. This has led to a completely new research area called Stochastic Differential Geometry, i.e. the stochastic analysis on manifolds. Stochastic methods open ways to solve analytic and geometric problems in a much more elegant fashion. Hence, we discuss briefly modern notions and key concepts of this relatively young and interdisciplinary subject.
Starting from the well-known notion of a flow to a vector field, we will show that stochastic flows naturally extend to a flow to a partial differential operator. As an application we sketch how these concepts can be used to give an elegant and short proof for existence and uniqueness of a solution to the Dirichlet problem.
This leads to the definition of Brownian motion on (smooth Riemannian) manifolds: the intrinsic approach as solution to the usual martingale problem using a Whitney embedding and the Eells-Elworthy-Malliavin approach using the projection from the orthonormal frame bundle.
Finally, we briefly discuss applications. In particular, curvature and gradient estimates of the heat semigroup.
Concerning the broad audience, all notions will be briefly introduced during the talk as needed.