Stochastic Lévy differential operators and Yang–Mills equations

Abstract

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