Finite difference calculus in the continuum

TL;DR

This paper explores finite difference calculus on configuration spaces, introducing new geometric insights, operator relations, and explicit Markov generators, advancing the mathematical framework for analyzing complex systems.

Contribution

It presents new results on finite difference geometry, operator relations, and explicit Markov generators on configuration spaces, expanding existing mathematical tools.

Findings

Finite difference geometry on configuration spaces is characterized.

Explicit forms for finite difference Markov generators are derived.

Connections between difference operators and canonical commutation relations are established.

Abstract

The paper describes known and new results about finite difference calculus on configuration spaces. We describe finite difference geometry on configuration spaces, connect finite difference operators with cannonical commutation relations, find explicit form for certain finite difference Markov generators on configuration spaces, and describe spaces of Newton series defined over the configuration spaces.