Open Access
October, 1986 Stochastic Determination of Moduli of Annular Regions and Tori
Hiroshi Yanagihara
Ann. Probab. 14(4): 1404-1410 (October, 1986). DOI: 10.1214/aop/1176992381


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.


Download Citation

Hiroshi Yanagihara. "Stochastic Determination of Moduli of Annular Regions and Tori." Ann. Probab. 14 (4) 1404 - 1410, October, 1986.


Published: October, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0606.30041
MathSciNet: MR866361
Digital Object Identifier: 10.1214/aop/1176992381

Primary: 30F20
Secondary: 58G32

Keywords: Brownian motion , modulus , Riemann surfaces , winding

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 4 • October, 1986
Back to Top