Polyharmonic Functions And Random Processes in Cones
- Mathematics of computing → Markov processes
- Theory of computation → Random walks and Markov chains
- Mathematics of computing → Enumeration
- Mathematics of computing → Generating functions
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.