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Absolute continuity of harmonic measure on rough domains

Murat Akman, University of Connecticut
Tuesday, May 9, 2017 - 1:30pm to 3:30pm
PDL C-401
Consider a Brownian particle moving in a domain \$Omega\$ in \$mathbb{R}^{n+1}\$. Using circular "detectors'' located on the boundary, our aim is to find the point where it first hits the boundary \$partial\Omega\$. Assume that a detector of radius \$r\$ costs us \$psi(r)\$ for some increasing function \$psi\$ on \$(0,\infty)\$, can we detect the exit point on a finite budget? 
In the first part of my talk, I will describe that how the detection question is equivalent for what functions \$psi\$, harmonic measure of \$Omega\$, \$omega_{\Omega}\$, is absolutely continuous with respect to \$psi-\$dimensional Hausdorff measure. I will provide some partial answers to this problem when \$Omega\$ has certain properties in the plane or in space by stating some historical results starting from a classical result of F. and M. Riesz. 
In the second part of my talk, we will study this problem when \$psi(r)=r^n\$ and we will try to find "minimal'' geometric or analytic assumptions on the domain \$Omega\$ to guarantee the absolute continuity of \$omega_{\Omega}\$ and \$n-\$dimensional Hausdorff measure.