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SUMMARY:New harmonic-measure distribution functions of simply connected an
 d multiply connected planar domains - Lesley Ward (University of South Aus
 tralia)
DTSTART:20230728T080000Z
DTEND:20230728T090000Z
UID:TALK202822@talks.cam.ac.uk
DESCRIPTION:We can encode information about the shape of the boundary of a
  planar region Ω by keeping track of the exit locations of Brownian part
 icles released from a basepoint z0 in Ω. Tracking for each r > 0 the pro
 bability that the Brownian particle first exits the region Ω via a bound
 ary point of Ω within the closed disc B(z0\, r) of radius r about z0 yie
 lds the (&ldquo\;global&rdquo\;) h-function or harmonic-measure distributi
 on function h(r) = hΩ\,z0(r). By contrast\, tracking the probability tha
 t the Brownian particle first exits the closed disc B(z0\, r) via a point 
 on the boundary &part\;Ω of Ω\, as opposed to via a point on &part\;B(
 z0\, r)\\&part\;Ω\, yields the (&ldquo\;local&rdquo\;) g-function g(r) =
  gΩ\,z0(r). I will report the recently calculated h-functions and g-func
 tions of several simply connected regions Ω\, as well as the h-functions
  of multiply connected domains whose boundary components may be circles\, 
 intervals or polygons. Tools used include conformal mapping\, properties o
 f harmonic measure\, and the prime function. This is joint work with Arunm
 aran Mahenthiram\, Byron Walden and Christopher Green.
LOCATION:Seminar Room 1\, Newton Institute
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