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Path signatures of ODE solutions

Francesco Galuppi, Giovanni Moreno, Pierpaola Santarsiero

TL;DR

The paper develops a geometric program to characterize solutions of ordinary differential equations through path signatures $\sigma(X)$, recasting ODE constraints in terms of algebraic conditions on the signature tensors. By embedding the problem in jet spaces and exploiting algebraic-variety and holonomic/Legendrian structures, it provides necessary and sufficient signature conditions for a path to solve a Cauchy problem, including explicit results for integral curves of linear and Hamiltonian vector fields. This approach enables reconstruction and verification of ODE solutions from signature data and highlights connections to Chen's uniqueness theorem, shuffle products, and holonomic jet extensions. The framework points to extensions to analytic manifolds, rough paths, and stochastic differential equations, and invites quantitative notions of proximity to varieties via partial signature constraints.

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.

Path signatures of ODE solutions

TL;DR

The paper develops a geometric program to characterize solutions of ordinary differential equations through path signatures , recasting ODE constraints in terms of algebraic conditions on the signature tensors. By embedding the problem in jet spaces and exploiting algebraic-variety and holonomic/Legendrian structures, it provides necessary and sufficient signature conditions for a path to solve a Cauchy problem, including explicit results for integral curves of linear and Hamiltonian vector fields. This approach enables reconstruction and verification of ODE solutions from signature data and highlights connections to Chen's uniqueness theorem, shuffle products, and holonomic jet extensions. The framework points to extensions to analytic manifolds, rough paths, and stochastic differential equations, and invites quantitative notions of proximity to varieties via partial signature constraints.

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.
Paper Structure (11 sections, 19 theorems, 95 equations, 1 figure)

This paper contains 11 sections, 19 theorems, 95 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Chen's uniqueness theorem and the shuffle identity
  3. Signatures of paths on an algebraic variety
  4. Signatures of holonomic paths
  5. Applications to ODE
  6. A geometric framework for systems of ODEs
  7. Characterization of solutions in terms of signature
  8. Integral curves of vector fields
  9. Linear vector fields
  10. Hamiltonian fields
  11. Conclusions and perspectives

Key Result

Theorem 2.3

Let $X,Y\in\mathcal{C}^{{1}\,\textrm{-pw}}([a,b])$ be paths. Then $\sigma(X)=\sigma(Y)$ if and only if $X_{red}$ and $Y_{red}$ are the same path, up to translation.

Figures (1)

  • Figure 1: \ref{['ex: path su cilindro']}.

Theorems & Definitions (59)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • ...and 49 more