# The Stochastic Energy-Casimir Method

## TL;DR

This paper extends the energy-Casimir stability method to stochastic Hamiltonian systems, providing probabilistic stability conditions for systems with symmetries and multiplicative noise, applicable to classical mechanical examples.

## Contribution

It introduces a stochastic extension of the energy-Casimir method, establishing probabilistic stability criteria for stochastic Lie-Poisson systems with noise.

## Key findings

- Stable deterministic equilibria remain stable in probability under stochastic perturbations.
- The stability in probability depends on the noise amplitude and system dimensionality.
- The theory is demonstrated on classical examples like the rigid body and Euler equations.

## Abstract

In this paper, we extend the energy-Casimir stability method for deterministic Lie-Poisson Hamiltonian systems to provide sufficient conditions for the stability in probability of stochastic dynamical systems with symmetries and multiplicative noise. We illustrate this theory with classical examples of coadjoint motion, including the rigid body, the heavy top and the compressible Euler equation in two dimensions. The main result of this extension is that stable deterministic equilibria remain stable in probability up to a certain stopping time which depends on the amplitude of the noise for finite dimensional systems and on the amplitude the spatial derivative of the noise for infinite dimensional systems.