Non linear gluon evolution in path-integral form

Abstract

We explore and clarify the connections between two different forms of the renormalisation group equations describing the quantum evolution of hadronic structure functions at small $x$. This connection is established via a Langevin formulation and associated path integral solutions that highlight the statistical nature of the quantum evolution, pictured here as a random walk in the space of Wilson lines. The results confirm known approximations, form the basis for numerical simulations and widen the scope for further analytical studies.