Least Constraint and Contact Dynamics of Stochastic Vector Bundles

TL;DR

This paper develops a geometric framework for stochastic vector bundles, introducing the least constraint theorem and contact dynamical equations, which help understand the evolution and constraints of stochastic systems.

Contribution

It introduces the least constraint theorem and contact dynamical equations for stochastic vector bundles, linking geometric structures to system evolution.

Findings

Stochastic vector bundles have an infinite-order jet structure.

They possess a natural contact structure that decomposes tangent spaces.

Derived contact dynamical equations represent the least constraint principle.

Abstract

This paper investigates the contact structures and dynamics of stochastic vector bundles, leading to the formulation of the least constraint theorem. It is found that the probability space of stochastic vector bundles possesses an infinite-order jet structure, which enables the geometric analysis of stochastic processes. Furthermore, this study demonstrates that stochastic vector bundles have a natural contact structure, leading to the decomposition of the tangent space and providing insight into the evolution and constraints of the system. Finally, we derive a set of contact dynamical equations for the stochastic vector bundles. These equations correspond to the least constraint on the evolution of stochastic vector bundles, which is a counterpart to the least action principle for symplectic structures. This shows the relationship between the geometric structure of the stochastic system evolution and its tendency to minimize constraints. This study provides a geometric framework for analyzing stochastic space with potential applications in various fields where probabilistic behavior is crucial.