Stochastic Determination of Moduli of Annular Regions and Tori

Abstract

Let $A = A(r, 1)$ be an annulus $\{z: r < |z| < 1\}$ with the Poincare metric $g$ on $A$. Let $\mathbf{Z} = (Z_t, P_a)$ be a Brownian motion on $A$ corresponding to $g$. If we take a geodesic disc $D$ centered at $c$ in $A$, then the probability $P_a(\exists t, Z_t \in \partial D$ such that $Z_s, 0 < s < t$, winds around the origin in the positive direction) is a function of $r, |c|$, and the radius $\rho$ of $D$. In the present paper we shall calculate the value $S$ of the supremum of these winding probabilities. Then it will turn out that there exists a 1 to 1 correspondence between $S$ and $r$. Noting that $r$ is called the modulus of $A$, we have an explicit formula of moduli of annular regions. Further we shall give an explicit formula of moduli of tori in a similar way.