On Stochastic Variational Principles

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.

19 pages

References in corpus (7)

• Variational Principles for Stochastic Fluid Dynamics
• Stochastic Variational Integrators
• Noise and dissipation on coadjoint orbits
• Semi-martingale driven variational principles
• Hamilton-Jacobi Theory for Degenerate Lagrangian Systems with Holonomic and Nonholonomic Constraints
• Momentum Maps and Stochastic Clebsch Action Principles
• From Second-Order Differential Geometry to Stochastic Geometric Mechanics