Hamiltonian stochastic Lie systems and applications

TL;DR

The paper addresses extending and correcting stochastic Lie systems by clarifying that the Stratonovich formulation preserves the classical Lie theorem, while the Itô form introduces drift that can break Lie-algebra closure. It develops Hamiltonian stochastic Lie systems, introduces stochastic foliated Lie systems, and extends the Poisson coalgebra method to derive superposition rules in the stochastic Hamiltonian setting. It also develops stability analysis, relative equilibria via symmetry reduction, and stochastic energy-momentum methods, with applications to SIS epidemic models, stochastic oscillators, and related biological and physical systems. Together, these results provide a practical, geometry-driven toolkit for analyzing stochastic Lie systems and their Hamiltonian structure across physics, biology, and beyond, including guidance on translating Itô models to Stratonovich form for Lie-theoretic methods.

Abstract

This paper provides a practical approach to stochastic Lie systems, i.e. stochastic differential equations whose general solutions can be written as a function depending only on a generic family of particular solutions and some constants related to initial conditions. We correct the stochastic Lie theorem characterising stochastic Lie systems, proving that, contrary to previous claims, it retains its classical form in the Stratonovich approach. Meanwhile, we show that the form of stochastic Lie systems may significantly differ from the classical one in the Itô formalism. New generalisations of stochastic Lie systems, like the so-called stochastic foliated Lie systems, are introduced. Subsequently, we focus on stochastic Lie systems that are Hamiltonian systems relative to different geometric structures. Special attention is paid to the symplectic case. We study their stability properties and lay the foundations of a stochastic energy-momentum method. A stochastic Poisson coalgebra method is developed to derive superposition rules for Hamiltonian stochastic Lie systems. Potential applications of our results are presented for biological stochastic models, stochastic oscillators, stochastic Lotka--Volterra systems, Palomba--Goodwin models, among others. Our findings complement previous approaches by using stochastic differential equations instead of deterministic equations designed to capture some of the features of models of stochastic nature.

Figures (2)

• Figure 1: These are two representations of the evolution in terms of the time of a particular solution to the Hamiltonian Lie system $\delta y=-x\delta t-Ax\delta W,\delta x=y\delta t+Ay\delta W$, with initial condition $(.2,0)$ a semi-martingale $W$, and a parameter $A$. The symplectic structure is $\omega=\mathrm{d} x\wedge \mathrm{d} y$. The system has a strong constant of motion $x^2+y^2$. The constant of motion is always conserved, but solutions jump back and forward relative to the deterministic solution.
• Figure 2: Representation of a solution to $\delta x=y\delta t-0.008\delta W,\delta y=-x\delta t-0.008\delta W$ on $\mathbb{R}^2$ with initial condition $(0,0)$ and stochastic variable given by a semi-martingale $W$, showing that a non-vanishing stochastic part of the stochastic differential equation may move solutions away from a deterministic equilibrium point.

Theorems & Definitions (26)

• Example 2.1
• Lemma 2.2: Itô–Stratonovich correction in coordinates
• Definition 2.3
• Example 2.4
• Example 2.5
• Definition 2.6
• Proposition 2.7
• Definition 2.8: Foliated stochastic Lie system
• Definition 3.1
• Theorem 3.2: Stochastic Lie theorem