• Mathematics of computing → Markov processes
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Generating functions
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

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.