Stochastic Levy Differential Operators and Yang-Mills Equations

TL;DR

This paper explores the connection between stochastic Levy differential operators and Yang-Mills equations, revealing differences from the deterministic case and deriving an equivalent stochastic equation involving Levy divergence.

Contribution

It introduces a stochastic Levy Laplacian and derives an equation equivalent to Yang-Mills equations using Cesaro averaging, highlighting new stochastic analytical tools.

Findings

Yang-Mills equations are not equivalent to Levy-Laplace equations in the stochastic setting

An equivalent stochastic Yang-Mills equation involving Levy divergence is obtained

Yang-Mills action functional is represented as an infinite-dimensional Dirichlet functional

Abstract

The relationship between the Yang-Mills equations and the stochastic analogue of Levy differential operators is studied. The value of the stochastic Levy Laplacian is found by means of Cesaro averaging of directional derivatives on the stochastic parallel transport. It is shown that the Yang-Mills equations and the Levy-Laplace equation for such Laplacian are not equivalent as in the deterministic case. An equation equivalent to the Yang-Mills equations is obtained. The equation contains the stochastic Levy divergence. It is proved that the Yang-Mills action functional can be represented as an infinite-dimensional analogue of the Direchlet functional of chiral field. This analogue is also derived using Cesaro averaging.