Nonlinear Dirichlet forms, energy spaces, and calculus rules

TL;DR

This paper reviews recent advances in nonlinear Dirichlet forms, focusing on 2-homogeneous local forms, and introduces new calculus-like properties inspired by Finsler geometry and metric measure spaces.

Contribution

It establishes novel properties of nonlinear Dirichlet forms that resemble differential calculus, inspired by Finsler manifolds and metric measure space theories.

Findings

New properties of nonlinear Dirichlet forms established

Connections to Finsler geometry and metric measure spaces

Enhanced understanding of calculus rules for energy spaces

Abstract

We review recent contributions on nonlinear Dirichlet forms. Then, we specialise to the case of 2-homogeneous and local forms. Inspired by the theory of Finsler manifolds and metric measure spaces, we establish new properties of such nonlinear Dirichlet forms, which are reminiscent of differential calculus formulae.