| Abstract: |
| Consider releasing a Brownian particle from a basepoint $z_{0}$ in a planar domain $\Omega \subset \mathbb{C}$. What is the chance, denoted $h_{\Omega,z_{0}}(r)$, that the particle`s first exit from $\Omega$ occurs within a fixed distance $r>0$ of $z_0$? The function of $r$ suggested by this question, $h_{\Omega,z_0}:[0,\infty)\rightarrow [0,1]$, is called the harmonic measure distribution function, or $h$-function, of $\Omega$ with respect to $z_{0}$. We can think of the $h$-function as a signature that encodes the geometry of the boundary of $\Omega$. In the language of PDEs, the $h$-function can also be formulated in terms of a suitable Dirichlet problem on $\Omega$. For simply connected domains, the theory of $h$-functions is now quite well developed and several explicit results are known. However, until recently, for multiply connected domains the theory of $h$-functions has been almost entirely out of reach.
We will show how to construct explicit formulae for the h-functions of symmetric multiply connected slit domains $\hat \Omega$ whose boundaries consist of an even number of colinear slits. We will employ the special function theory of the Schottky-Klein prime function $\omega(\zeta,\gamma)$ and its associated constructive methods in conformal mapping to build explicit formulae for the h-functions of domains $\hat \Omega$ with any finite even number of slits. |
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