Ellen Krusell: The Dirichlet Problem through Brownian Motion

As we shall see in the colloquium talk, some problems from complex analysis, which are seemingly unrelated to probability theory, have a fruitful interpretation in terms of Brownian motion. In this talk, we will try to gain some intuition for why this is the case by considering a basic example of this phenomenon, namely, the Dirichlet problem. We will first recall some basic properties of Brownian motion, conformal maps, and harmonic functions. Then, we will have a look at the Dirichlet problem and, the closely related, harmonic measure. We will see that the latter can be interpreted as a hitting probability of Brownian motion. This will lead us to a solution of the Dirichlet problem in terms of Brownian motion.

This talk precedes Yilin Wang's colloquium talk  on “The Brownian loop measure on Riemann surfaces, length spectra, and determinants of Laplacians” and is accessible to a broad audience (master and PhD students).