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The harmonic-measure distribution function of a planar domain, and the Schottky-Klein prime function

Presented by: 
Lesley Ward University of South Australia
Date: 
Friday 13th September 2019 - 10:00 to 11:00
Venue: 
INI Seminar Room 1
Abstract: 
The $h$-function or harmonic-measure distribution function $h(r) = h_{\Omega, z_0}(r)$ of a planar region $\Omega$ with respect to a basepoint $z_0$ in $\Omega$ records the probability that a Brownian particle released from $z_0$ first exits $\Omega$ within distance $r$ of $z_0$, for $r > 0$. For simply connected domains $\Omega$ the theory of $h$-functions is now well developed, and in particular the $h$-function can often be computed explicitly, making use of the Riemann mapping theorem. However, for multiply connected domains the theory of $h$-functions has been almost entirely out of reach. I will describe recent work showing how the Schottky-Klein prime function $\omega(\zeta,\alpha)$ allows us to compute the $h$-function explicitly, for a model class of multiply connected domains. This is joint work with Darren Crowdy, Christopher Green, and Marie Snipes.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons