Brownian Motion, Quasiconformal Mappings and the Beltrami Equation

Abstract

Consider a Jordan domain $\Omega$ in the plane with $3$ distinct points marked on its boundary. These $3$ points split $\partial \Omega$ into $3$ arcs. For each $z \in \Omega$, we can assign it the harmonic coordinates by taking the harmonic measures from $z$ to each of these $3$ arcs on the boundary. We showed that these harmonic coordinates uniquely characterize the interior points of $\Omega$. In particular, we can define these harmonic coordinates for points in the unit disk $\mathbb{D}$ with $3$ distinct points marked on its boundary. We further showed that we can define a map $\phi$ from $\Omega$ onto $\mathbb{D}$ by sending each point in $\Omega$ to the point in $\mathbb{D}$ with the same harmonic coordinates, and this map is one-to-one and analytic. We also gave an elementary proof of the conformal invariance of Brownian motion, and since harmonic measure can be defined in terms of Brownian motion, this givesus a new perspective on the Riemann mapping theorem. Furthermore, this strategy could potentially be generalized to prove the measurable Riemann mapping theorem. In this case, the map $\phi$ from $\Omega$ to $\mathbb{D}$ is assumed to be a quasiconformal map satisfying the Beltrami equation $\phi_{\overline{z}} = \mu(z) \phi_z$ almost everywhere for certain Beltrami coefficient $\mu: \Omega \rightarrow \mathbb{C}$ measurable with $||\mu||_\infty \le k < 1$. However, a Brownian motion is in general not preserved under quasiconformal maps. In order to define the correct coordinates for this strategy to work, it is necessary to consider the harmonic measure of stochastic processes which get mapped to a Brownian motion under quasiconformal maps. In this direction, we proved that we can explicitly construct the stochastic processes (up to a time-change) whose images under certain quasiconformal maps are Brownian motions. Moreover, this construction requires only information about the Beltrami coefficient.