On Stochastic Variational Principles

2025/04/08 by Saha, Archishman · 1 citation
#70G75 #70H30 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2504.06411

Abstract

The study of stochastic variational principles involves the problem of constructing fixed-endpoint and adapted variations of semimartingales. We provide a detailed construction of variations of semimartingales that are not only fixed at deterministic endpoints, but also fixed at first entry times and first exit times for charts in a manifold. We prove a stochastic version of the fundamental lemma of calculus of variations in the context of these variations. Using this framework, we provide a generalization of the stochastic Hamilton-Pontryagin principle in local coordinates to arbitrary noise semimartingales. For the corresponding global form of the stochastic Hamilton-Pontryagin principle, we introduce a novel approach to global variational principles by restricting to semimartingales obtained as solutions of Stratonovich equations.

Citations

• Semimartingale driven mechanics and reduction by symmetry for stochastic and dissipative dynamical systems
• Semi-martingale driven variational principles
• Momentum Maps and Stochastic Clebsch Action Principles
• Noise and dissipation on coadjoint orbits
• Variational Principles for Stochastic Fluid Dynamics
• The research program of Stochastic Deformation (with a view toward Geometric Mechanics)
• Hamilton–Jacobi theory for degenerate Lagrangian systems with holonomic and nonholonomic constraints
• Lagrangian Navier-Stokes diffusions on manifolds: variational principle and stability
• The stochastic Hamilton-Jacobi equation
• Stochastic Variational Integrators
• Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations
• Stochastic Hamiltonian dynamical systems
• Dynamical Theories of Brownian Motion
• Derivation of the Schrödinger Equation from Newtonian Mechanics

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• Stochastic Hamiltonian dynamical systems
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• Lemme de coherence et théorème de Noether stochastique
• A note on invariant manifolds for stochastic partial differential equations in the framework of the variational approach