Contact/Symplectic Structure of Stochastic Vector Bundles

D. Y. Zhong, G. Q. Wang

Published 2024-08-21Version 1

In this study, we present an exploration of the geometric characteristics of stochastic vector bundles. The paper begins by introducing the jet bundle emanating from the probability space of vector bundles. Subsequently, we derive the contact form of this jet bundle. The central contribution of this work is the derivation of the Hamilton equation for the jet space, which reveals that the jet space of the probability space exhibits a contact structure. Notably, this structure reduces to a symplectic manifold in the context of autonomous systems. This reduction leads to a system equation that parallels the Hamilton equation, with the notable distinction that the Hamiltonian is substituted by the derivative of the probability. Our findings suggest the potential for a new geometric approach to addressing stochastic systems, offering a framework that may enhance our understanding and manipulation of these systems.