Non linear gluon evolution in path-integral form

We explore and clarify the connections between two different forms of the renormalization 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. (C) 2002 Elsevier Science B.V. All rights reserved.