Applications of standard and Hamiltonian stochastic Lie systems

TL;DR

This paper explores stochastic Lie systems and Hamiltonian stochastic Lie systems, providing new examples, extending the coalgebra method, and applying the theory to various biological, physical, and mathematical models.

Contribution

It introduces new stochastic Lie system examples, extends the coalgebra method for Hamiltonian cases, and applies these theories to diverse real-world models.

Findings

New stochastic Lie system examples provided

Extended coalgebra method for Hamiltonian systems

Applied theory to biological and physical models

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.