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Harmonic measures

Jonatan Vasilis (Institutionen för matematiska vetenskaper, matematik)
Göteborg : Chalmers University of Technology, 2010. ISBN: 978-91-7385-423-8.- xii+185 s.
[Doktorsavhandling]

This thesis uses both analytic and probabilistic methods to study continuous and discrete problems. The main areas of study are the asymptotic properties of p-harmonic measure, and various aspects of the square root of the Poisson kernel.

Fix a domain and a boundary point, subject to certain regularity conditions. Consider the part of the boundary that lies within a disc, centered at the fixed boundary point. It is shown that as the radius of the disc tends to zero, the p-harmonic measure of the boundary set decays as an explicitly given power of the radius.

The square root of the Poisson kernel is studied in both continuous and discrete settings. In the continuous case the domain is the unit disc, and a Hardy space related to the square root of the Poisson kernel is defined. The main result is that, as opposed to the classical Hardy space, the positive functions do not admit a characterization in terms of an Orlicz space. Similar results are given also in the discrete case, where the domain is instead a regular tree.

Further results in the discrete setting include the construction of a nearest neighbor random walk on the tree with exit distribution determined by powers of the Poisson kernel. The minimally thin sets of these random walks are characterized.

Finally, we suggest a generalization of a two-dimensional geometric result – the ring lemma – to three dimensions.

Nyckelord: harmonic measure, square root of the Poisson kernel, Hardy space, p-Laplace operator, random walk, regular tree, discrete minimal thinness



Denna post skapades 2010-09-07. Senast ändrad 2013-09-25.
CPL Pubid: 125758

 

Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, matematik (2005-2016)

Ämnesområden

Matematisk analys
Diskret matematik
Annan matematik

Chalmers infrastruktur

Examination

Datum: 2010-10-07
Tid: 13:15
Lokal: sal Euler, Skeppsgränd 3, Göteborg
Opponent: professor Pekka Koskela, University of Jyväskylä, Finland

Ingår i serie

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie 3104