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Polyharmonic Functions And Random Processes in Cones

Authors: François Chapon, Éric Fusy, and Kilian Raschel

Published in: LIPIcs, Volume 159, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)


Abstract
We investigate polyharmonic functions associated to Brownian motions and random walks in cones. These are functions which cancel some power of the usual Laplacian in the continuous setting and of the discrete Laplacian in the discrete setting. We show that polyharmonic functions naturally appear while considering asymptotic expansions of the heat kernel in the Brownian case and in lattice walk enumeration problems. We provide a method to construct general polyharmonic functions through Laplace transforms and generating functions in the continuous and discrete cases, respectively. This is done by using a functional equation approach.

Cite as

François Chapon, Éric Fusy, and Kilian Raschel. Polyharmonic Functions And Random Processes in Cones. In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{chapon_et_al:LIPIcs.AofA.2020.9,
  author =	{Chapon, Fran\c{c}ois and Fusy, \'{E}ric and Raschel, Kilian},
  title =	{{Polyharmonic Functions And Random Processes in Cones}},
  booktitle =	{31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)},
  pages =	{9:1--9:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-147-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{159},
  editor =	{Drmota, Michael and Heuberger, Clemens},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2020.9},
  URN =		{urn:nbn:de:0030-drops-120390},
  doi =		{10.4230/LIPIcs.AofA.2020.9},
  annote =	{Keywords: Brownian motion in cones, Heat kernel, Random walks in cones, Harmonic functions, Polyharmonic functions, Complete asymptotic expansions, Functional equations}
}
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