Kinetic Equation for Stochastic Vector Bundles

TL;DR

This paper introduces a novel kinetic equation for stochastic systems on vector bundles, capturing global, non-Markovian effects and extending classical Fokker-Planck analysis to a broader, more complex setting.

Contribution

It derives a new kinetic equation based on cumulant expansion that addresses global randomness and non-Markovianity in stochastic vector bundle systems.

Findings

The kinetic equation reduces to the classical Fokker-Planck for Markovian processes.

It captures global and historical influences in stochastic dynamics.

Discusses limitations and potential approximations of the kinetic equation.

Abstract

The kinetic equation is crucial for understanding the statistical properties of stochastic processes, yet current equations, such as the classical Fokker-Planck, are limited to local analysis. This paper derives a new kinetic equation for stochastic systems on vector bundles, addressing global scale randomness. The kinetic equation was derived by cumulant expansion of the ensemble-averaged local probability density function, which is a functional of state transition trajectories. The kinetic equation is the geodesic equation for the probability space. It captures global and historical influences, accounts for non-Markovianity, and can be reduced to the classical Fokker-Planck equation for Markovian processes. This paper also discusses relative issues concerning the kinetic equation, including non-Markovianity, Markov approximation, macroscopic conservation equations, gauge transformation, and truncation of the infinite-order kinetic equation, as well as limitations that require further attention.