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Applications of standard and Hamiltonian stochastic Lie systems

Javier de Lucas, Marcin Zając

TL;DR

The paper develops stochastic Lie system theory, including Hamiltonian variants, by extending the Stratonovich operator framework and the VG Lie algebra structure. It advances the Poisson coalgebra method to SDEs with coefficients depending on stochastic parameters, enabling explicit superposition rules. Key contributions include new superposition formulas for stochastic Riccati, Ermakov, and Lotka–Volterra systems, as well as applications to coronavirus models, demonstrating Hamiltonian structures and diagonal prolongations. These results broaden the applicability of stochastic Lie systems to biological and epidemiological dynamics and point to further extensions such as stochastic matrix Riccati equations.

Abstract

A stochastic Lie system on a manifold $M$ is a stochastic differential equation whose dynamics is described by a linear combination with functions depending on $\mathbb{R}^\ell$-valued semi-martigales of vector fields on $M$ spanning a finite-dimensional Lie algebra. We analyse new examples of stochastic Lie systems and Hamiltonian stochastic Lie systems, and review and extend the coalgebra method for Hamiltonian stochastic Lie systems. We apply the theory to biological and epidemiological models, stochastic oscillators, stochastic Riccati equations, coronavirus models, stochastic Ermakov systems, etc.

Applications of standard and Hamiltonian stochastic Lie systems

TL;DR

The paper develops stochastic Lie system theory, including Hamiltonian variants, by extending the Stratonovich operator framework and the VG Lie algebra structure. It advances the Poisson coalgebra method to SDEs with coefficients depending on stochastic parameters, enabling explicit superposition rules. Key contributions include new superposition formulas for stochastic Riccati, Ermakov, and Lotka–Volterra systems, as well as applications to coronavirus models, demonstrating Hamiltonian structures and diagonal prolongations. These results broaden the applicability of stochastic Lie systems to biological and epidemiological dynamics and point to further extensions such as stochastic matrix Riccati equations.

Abstract

A stochastic Lie system on a manifold is a stochastic differential equation whose dynamics is described by a linear combination with functions depending on -valued semi-martigales of vector fields on spanning a finite-dimensional Lie algebra. We analyse new examples of stochastic Lie systems and Hamiltonian stochastic Lie systems, and review and extend the coalgebra method for Hamiltonian stochastic Lie systems. We apply the theory to biological and epidemiological models, stochastic oscillators, stochastic Riccati equations, coronavirus models, stochastic Ermakov systems, etc.

Paper Structure

This paper contains 5 sections, 1 theorem, 32 equations.

Table of Contents

  1. Introduction
  2. Superposition rules and Poisson coalgebra method
  3. New applications of stochastic Lie systems
  4. Conclusions
  5. Acknowledgments

Key Result

proposition 1

If $\mathfrak{H}$ is a HS Lie system with an $\mathbb{R}^\ell$-dependent Hamiltonian $\widetilde{h}:\mathbb{R}^\ell \times M\rightarrow \mathbb{R}^\ell$ relative to a symplectic form $\omega$ on $M$, then $\mathfrak{H}^{[m]}$ is a HS Lie system relative to $\omega^{[m]}$ with a Hamiltonian $\widetil

Theorems & Definitions (4)

  • definition 1
  • definition 2
  • definition 3
  • proposition 1