Kinetic Equation for Stochastic Vector Bundles

The kinetic equation is the conservation equation for the probability density function (PDF) of stochastic systems, which is essential in understanding statistical properties of stochastic processes found in both natural and social phenomena. However, currently reported kinetic equations, for instance, the classical Fokker-Planck equation, are usually limited to local stochastic analysis, making them inadequate in exploring the overall characteristics of random processes on a global scale. This paper derived a kinetic equation for stochastic systems defined on vector bundles. The stochastic vector bundles are constructed by embedding a probability space in a configuration manifold. The local PDF and the cumulant expansion method are introduced in the formulation. The local PDF is assumed to be a function of state transition trajectories, geometrically understood as sections of the stochastic vector bundle. Two features characterize the kinetic equation. Firstly, it shows to be the geodesic equation of the prolonged probability space, which is intrinsic and coordinate-independent. Secondly, the coefficients of the kinetic equation are the functions of cumulants for the integrals of tangent vectors to stochastic vector fibers. These two characteristics make the new kinetic equation capable of considering both global and historical effects on stochastic processes. Non-Markovinity and Markovian approximations of the kinetic equation are also discussed. It shows that the kinetic equation can apply to global non-Markovian processes, and it reduces to the classical Fokker-Planck equation for Markovian processes defined on a flat Euclidean space.