Path signatures of ODE solutions

TL;DR

This paper characterizes the signature tensors of paths that are solutions to specific ODE systems, providing algebraic conditions and applications to linear and Hamiltonian systems.

Contribution

It establishes necessary and sufficient algebraic conditions for path signatures to correspond to solutions of ODEs, advancing the theoretical understanding of path signatures.

Findings

Derived algebraic conditions for ODE solution signatures

Applied conditions to linear and Hamiltonian systems

Enhanced the theoretical framework for path signatures in differential equations

Abstract

The signature of a path is a sequence of tensors which allows to uniquely reconstruct the path. By employing the geometric theory of nonlinear systems of ordinary differential equations, we find necessary and sufficient algebraic conditions on the signature tensors of a path to be a solution of a given system of ODEs. As an application, we describe in detail the systems of ODEs that describe the trajectories of a vector field, in particular a linear and Hamiltonian one.