Stochastic determination of moduli of annular regions and tori.(English)Zbl 0606.30041

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.

MSC:

 30F20 Classification theory of Riemann surfaces 58J65 Diffusion processes and stochastic analysis on manifolds
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