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The stochastic Hamilton-Jacobi-Bellman equation on Jacobi structures

Pingyuan Wei, Qiao Huang, Jinqiao Duan

TL;DR

This work extends stochastic Hamiltonian mechanics to the broad setting of Jacobi manifolds, unifying Poisson, contact, and locally conformal symplectic (L.C.S.) geometries under stochastic dynamics. It develops a rigorous stochastic Hamiltonian framework in Stratonovich and Itô forms and proves preservation of characteristic leaves and geometric structures by stochastic flows, along with a stochastic Liouville-type theorem. The stochastic Hamilton–Jacobi–Bellman (HJB) theory is developed for both contact and LC.S cases, presenting two formalisms for generating functions and establishing conditions for complete stochastic solutions and their first-integrals. These results connect stochastic control perspectives with geometric mechanics on Jacobi manifolds, enabling analysis of dissipative and noisy systems within a unified geometric framework and offering new avenues for stochastic reduction and integration in non-Poisson settings.

Abstract

Jacobi structures are known to generalize Poisson structures, encompassing symplectic, cosymplectic, and Lie-Poisson manifolds. Notably, other intriguing geometric structures -- such as contact and locally conformal symplectic manifolds -- also admit Jacobi structures but do not belong to the Poisson category. In this paper, we employ global stochastic analysis techniques, initially developed by Meyer and Schwartz, to rigorously introduce stochastic Hamiltonian systems on Jacobi manifolds. We then propose a stochastic Hamilton-Jacobi-Bellman (HJB) framework as an alternative perspective on the underlying dynamics. We emphasize that many of our results extend the work of Bismut [Bis80, Bis81] and Lázaro-Camí \& Ortega [LCO08, LCO09]. Furthermore, aspects of our geometric Hamilton-Jacobi theory in the stochastic setting draw inspiration from the deterministic contributions of Abraham \& Marsden [AM78], de León \& Sardón [dLS17], Esen et al. [EdLSZ21], and related literature.

The stochastic Hamilton-Jacobi-Bellman equation on Jacobi structures

TL;DR

This work extends stochastic Hamiltonian mechanics to the broad setting of Jacobi manifolds, unifying Poisson, contact, and locally conformal symplectic (L.C.S.) geometries under stochastic dynamics. It develops a rigorous stochastic Hamiltonian framework in Stratonovich and Itô forms and proves preservation of characteristic leaves and geometric structures by stochastic flows, along with a stochastic Liouville-type theorem. The stochastic Hamilton–Jacobi–Bellman (HJB) theory is developed for both contact and LC.S cases, presenting two formalisms for generating functions and establishing conditions for complete stochastic solutions and their first-integrals. These results connect stochastic control perspectives with geometric mechanics on Jacobi manifolds, enabling analysis of dissipative and noisy systems within a unified geometric framework and offering new avenues for stochastic reduction and integration in non-Poisson settings.

Abstract

Jacobi structures are known to generalize Poisson structures, encompassing symplectic, cosymplectic, and Lie-Poisson manifolds. Notably, other intriguing geometric structures -- such as contact and locally conformal symplectic manifolds -- also admit Jacobi structures but do not belong to the Poisson category. In this paper, we employ global stochastic analysis techniques, initially developed by Meyer and Schwartz, to rigorously introduce stochastic Hamiltonian systems on Jacobi manifolds. We then propose a stochastic Hamilton-Jacobi-Bellman (HJB) framework as an alternative perspective on the underlying dynamics. We emphasize that many of our results extend the work of Bismut [Bis80, Bis81] and Lázaro-Camí \& Ortega [LCO08, LCO09]. Furthermore, aspects of our geometric Hamilton-Jacobi theory in the stochastic setting draw inspiration from the deterministic contributions of Abraham \& Marsden [AM78], de León \& Sardón [dLS17], Esen et al. [EdLSZ21], and related literature.

Paper Structure

This paper contains 17 sections, 18 theorems, 152 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Jacobi structures & manifolds
  4. Hamiltonian formulations
  5. Characteristic foliations and Lagrangian--Legendrian submanifolds
  6. Stochastic calculus on manifolds
  7. Stochastic Hamiltonian Systems (SHSs) on Jacobi Manifolds
  8. Definitions of SHSs in Stratonovich and Itô forms
  9. Preservation of characteristic leaves $\&$ structures
  10. Examples
  11. The stochastic Hamilton--Jacobi--Bellman (HJB) theory
  12. Stochastic HJ equations for contact cases
  13. A contact HJB formalism on $Q$
  14. An alternative formalism on $Q\times \mathbb{R}$
  15. Stochastic Hamilton--Jacobi equations for the L.C.S. case
  16. ...and 2 more sections

Key Result

Proposition 2.3

Let $(M,\Lambda,E)$ be a Jacobi manifold and $f\in C^\infty (M)$. The smooth curve $\gamma: [0,T] \to M$ is an integral curve of the Hamiltonian vector field $V_f$ if and only if for any $\alpha\in\Omega^1(M)$ and for any $t \in [0,T]$, or equivalently,

Figures (1)

  • Figure 1: Jacobi manifolds (see Table 1 in Appendix D for details of the notations).

Theorems & Definitions (55)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 45 more