A Laplacian on the Full Shift Space

Abstract

In this paper, we consider the one-sided shift space on finitely many symbols and extend the theory of what is known as rough analysis. We define difference operators on an increasing sequence of subsets of the shift space that would eventually render the Laplacian on the space of real-valued continuous functions on the shift space. We then define the Green's function and the Green's operator that come in handy to solve the analogue to the Dirichlet boundary value problem on the shift space.