On the Attracting Orbit of a Nonlinear Transformation Arising from Renormalization of Hierarchically Interacting Diffusions

This paper analyzes then-fold composition of a non-linear integral operator acting on a class of functions on [0, ∞). Attracting orbits and attracting fixed points are identified. Various results of convergence to these orbits and to these fixed points are derived. The proofs are based on order-preserving properties and comparison techniques. A key role is played by the eigenfunctions of the operator, which are used as comparison objects. The results imply that the space-time scaling limit of an infinite system of interacting diffusions has universal behavior independent of model parameters. The paper can be read independently of Part I.