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Polyharmonic Functions And Random Processes in Cones

Authors François Chapon, Éric Fusy, Kilian Raschel

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Author Details

François Chapon
  • Université de Toulouse, Institut de Mathématiques de Toulouse, UMR CNRS 5219, 31062 Toulouse Cedex 9, France
Éric Fusy
  • CNRS, LIX, UMR CNRS 7161, École Polytechnique, 1 rue Honoré d'Estienne d'Orves, 91120 Palaiseau, France
Kilian Raschel
  • CNRS, Institut Denis Poisson, UMR CNRS 7013, Université de Tours et Université d'Orléans, Parc de Grandmont, 37200 Tours, France

Funding

  • Chapon, François: F. C. is supported by the grant ANR-18-CE40-0006 MESA funded by the French National Research Agency (ANR).
  • Fusy, Éric: É. F. is supported by the grant ANR-16-CE40-0009-01 GATO funded by the French National Research Agency (ANR).
  • Raschel, Kilian: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under the Grant Agreement No. 759702.

Acknowledgements

We would like to thank Cédric Lecouvey, Steve Melczer, Pierre Tarrago and Wolfgang Woess for very interesting discussions. This project has started in July 2019, when two authors were invited at the Institute of Mathematical Statistics of Münster University. The institute, and in particular Gerold Alsmeyer, is greatly acknowledged for hospitality. The first author also acknowledges the Institut Denis Poisson for the warm hospitality during his stay at the Université de Tours, where part of this work has been pursued. Finally, we thank the three anonymous referees for useful comments.

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.AofA.2020.9

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Markov processes
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Generating functions
Keywords
  • Brownian motion in cones
  • Heat kernel
  • Random walks in cones
  • Harmonic functions
  • Polyharmonic functions
  • Complete asymptotic expansions
  • Functional equations

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