[Paper Review] Stochastic Hamiltonian dynamical systems

Joan-Andreu Lázaro-Camí, Juan‐Pablo Ortega|ArXiv.org|Feb 26, 2007

Numerical methods for differential equations10 references4 citations

TL;DR

This paper formulates a stochastic generalization of Hamiltonian dynamics on Poisson manifolds using global stochastic analysis tools, introducing a critical action principle that fully characterizes stochastic Hamiltonian equations. It extends classical variational principles to include continuous semimartingale noise and establishes conditions for stability and conservation laws in the stochastic setting.

ABSTRACT

We use the global stochastic analysis tools introduced by P. A. Meyer and L. Schwartz to write down a stochastic generalization of the Hamilton equations on a Poisson manifold that, for exact symplectic manifolds, are characterized by a natural critical action principle similar to the one encountered in classical mechanics. Several features and examples in relation with the solution semimartingales of these equations are presented.

Motivation & Objective

• To develop a geometric framework for stochastic Hamiltonian systems on non-Euclidean phase spaces using global stochastic analysis.
• To generalize the classical variational principle to stochastic settings, fully characterizing stochastic Hamiltonian equations.
• To establish criteria for stability (almost sure and in probability) using stochastic conservation laws and energy methods.
• To model complex systems with inherent randomness, such as damped oscillators and Brownian motion on manifolds, via stochastic Hamiltonian dynamics.
• To unify stochastic mechanics with geometric mechanics by extending tools to Poisson and symplectic manifolds with general semimartingale noise.

Proposed method

• Uses P. A. Meyer and L. Schwartz's global stochastic analysis framework to handle non-Euclidean phase spaces.
• Models stochastic dynamics via continuous semimartingales, not restricted to Brownian motion, enabling broader applicability.
• Defines stochastic Hamiltonian equations on Poisson manifolds using Stratonovich and Itô stochastic differential equations with appropriate operators.
• Introduces a stochastic variational principle (Theorem 4.14) that fully characterizes the stochastic Hamiltonian equations, generalizing the deterministic critical action principle.
• Applies the stochastic Dirichlet criterion (Theorem 2.15) to analyze stability using two notions of conserved quantities: one for almost sure stability and one for stability in probability.
• Relies on the equivalence between Stratonovich and Itô formulations via a transformation between Stratonovich and Schwartz operators on tangent and second-order tangent bundles.

Experimental results

Research questions

• RQ1 How can the classical Hamiltonian formalism be generalized to include stochastic perturbations on Poisson manifolds?
• RQ2 What is the appropriate stochastic variational principle that fully characterizes stochastic Hamiltonian dynamics?
• RQ3 How do conserved quantities in the stochastic setting relate to stability properties such as almost sure Lyapunov stability and stability in probability?
• RQ4 Can stochastic Hamiltonian systems model physical phenomena like damping and Brownian motion on manifolds?
• RQ5 How do global stochastic analysis tools enable the reduction of stochastic Hamiltonian systems under symmetry?

Key findings

• The stochastic Hamiltonian equations on Poisson manifolds are fully characterized by a critical action principle, generalizing the deterministic case.
• The solution semimartingales preserve symplectic leaves, ensuring geometric consistency in the stochastic evolution.
• Two distinct notions of conserved quantities in the stochastic setting allow for the application of a stochastic Dirichlet criterion to conclude almost sure Lyapunov stability and stability in probability.
• A damped oscillator can be described as the average motion of the solution semimartingale of a stochastic Hamiltonian system, linking stochastic dynamics to macroscopic damping.
• Brownian motion on a manifold arises as the projection of a simple Hamiltonian semimartingale defined on the cotangent bundle or orthonormal frame bundle.
• The equivalence between Stratonovich and Itô formulations is established via a transformation between Stratonovich and Schwartz operators, ensuring consistency in stochastic modeling.