From Second-Order Differential Geometry to Stochastic Geometric Mechanics

TL;DR

This paper extends classical geometric mechanics into the stochastic domain using second-order differential geometry, establishing new structures and symmetries for stochastic differential equations and their relation to HJB equations.

Contribution

It introduces second-order geometric structures for stochastic mechanics, enabling the study of symmetries, variational principles, and Hamilton-Jacobi theory in stochastic systems.

Findings

Derived stochastic prolongation formulae for SDE symmetries

Established stochastic Hamilton's equations from second-order symplectic structures

Proved a stochastic Noether's theorem and linked to optimal transport and quantum mechanics

Abstract

Classical geometric mechanics, including the study of symmetries, Lagrangian and Hamiltonian mechanics, and the Hamilton-Jacobi theory, are founded on geometric structures such as jets, symplectic and contact ones. In this paper, we shall use a partly forgotten framework of second-order (or stochastic) differential geometry, developed originally by L. Schwartz and P.-A. Meyer, to construct second-order counterparts of those classical structures. These will allow us to study symmetries of stochastic differential equations (SDEs), to establish stochastic Lagrangian and Hamiltonian mechanics and their key relations with second-order Hamilton-Jacobi-Bellman (HJB) equations. Indeed, stochastic prolongation formulae will be derived to study symmetries of SDEs and mixed-order Cartan symmetries. Stochastic Hamilton's equations will follow from a second-order symplectic structure and canonical transformations will lead to the HJB equation. A stochastic variational problem on Riemannian manifolds will provide a stochastic Euler-Lagrange equation compatible with HJB one and equivalent to the Riemannian version of stochastic Hamilton's equations. A stochastic Noether's theorem will also follow. The inspirational example, along the paper, will be the rich dynamical structure of Schr\"odinger's problem in optimal transport, where the latter is also regarded as a Euclidean version of hydrodynamical interpretation of quantum mechanics.