The motion of a random string

TL;DR

This paper discusses the development of a natural evolution process on loops in Riemannian manifolds, clarifying algebraic structures related to renormalization in regularity structures.

Contribution

It introduces a new evolution on loop spaces and elucidates the algebraic aspects of renormalization procedures in regularity structures.

Findings

Construction of a natural evolution on loop spaces

Clarification of algebraic structures in renormalization

Forthcoming results in the theory of regularity structures

Abstract

We review a series of forthcoming results leading to the construction of a natural evolution on the space of loops with values in a Riemannian manifold. In particular, this clarifies the algebraic structure of the renormalisation procedures appearing in the context of the theory of regularity structures.