Polyharmonic Functions in the Quarter Plane (Extended Abstract)

TL;DR

This paper introduces a new method to compute all discrete and continuous polyharmonic functions in the quarter plane for models with small steps, zero drift, and finite group, and demonstrates convergence between the two cases.

Contribution

It presents a novel unified approach for discrete and continuous polyharmonic functions in the quarter plane, including convergence analysis.

Findings

New method for discrete polyharmonic functions in the quarter plane

Extension of method to continuous polyharmonic functions

Proof of convergence between discrete and continuous cases

Abstract

In this article, a novel method to compute all discrete polyharmonic functions in the quarter plane for models with small steps, zero drift and a finite group is proposed. A similar method is then introduced for continuous polyharmonic functions, and convergence between the discrete and continuous cases is shown.