Polyharmonic functions and random processes in cones

Francois Chapon (1) , Eric Fusy (2, 3) , Kilian Raschel (4, 3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
3 CNRS - Centre National de la Recherche Scientifique
4 IDP - Institut Denis Poisson

We investigate polyharmonic functions associated to Brownian motion 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.

Mots clés: Brownian motion in cones; Heat kernel; Random walks in cones; Harmonic functions; Polyharmonic functions; Complete asymptotic expansions; Functional equations

Domaines: Probabilités [math.PR]; Combinatoire [math.CO]

Francois Chapon, Eric Fusy, Kilian Raschel. Polyharmonic functions and random processes in cones. 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020), Clemens Heuberger, Jun 2020, Klagenfurt, Austria. pp.9:1--9:19. ⟨hal-02436386v2⟩