Stochastic determination of moduli of annular regions and tori (Q1084550)

Let \(A=A(r,1)\) be an annulus \(\{\) \(z: r<| z| <1\}\) with the Poincaré metric g on A. Let \({\mathbb{Z}}=(Z_ t,P_{\alpha})\) be a Brownian motion on A corresponding to g. If we take a geodesic disc D centered at c in A, then the probability \(P_{\alpha}\) (\(\exists t\), \(Z_ t\in \partial D\) 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.