Weak Formulation of the Laplacian on the Full Shift Space

TL;DR

This paper develops a weak formulation of the Laplacian on the full shift space, establishing fundamental analytical relations and boundary conditions, extending classical calculus concepts to symbolic dynamical systems.

Contribution

It introduces a weak Laplacian definition on the shift space, connecting it with energy forms and Neumann derivatives, and provides conditions for boundary value problems.

Findings

Defined a weak Laplacian on the shift space.

Established a Gauss-Green formula in this setting.

Provided conditions for Neumann boundary value problems.

Abstract

We consider a Laplacian on the one-sided full shift space over a finite symbol set, which is constructed as a renormalized limit of finite difference operators. We propose a weak definition of this Laplacian, analogous to the one in calculus, by choosing test functions as those which have finite energy and vanish on various boundary sets. In the abstract setting of the shift space, the boundary sets are chosen to be the sets on which the finite difference operators are defined. We then define the Neumann derivative of functions on these boundary sets and establish a relation between three important concepts in analysis so far, namely, the Laplacian, the bilinear energy form and the Neumann derivative of a function. As a result, we obtain the Gauss-Green's formula analogous to the one in classical case. We conclude this paper by providing a sufficient condition for the Neumann boundary value problem on the shift space.