Renormalization of Hierarchically Interacting Isotropic Diffusions

Abstract

We study a renormalization transformation arising in an infinite system of interacting diffusions. The components of the system are labeled by the N-dimensional hierarchical lattice (N≥2) and take values in the closure of a compact convex set $\bar D \subset \mathbb{R}^d (d \geqslant 1)$. Each component starts at some θ ∈ D and is subject to two motions: (1) an isotropic diffusion according to a local diffusion rate g: $\bar D \to [0,\infty ]$ chosen from an appropriate class; (2) a linear drift toward an average of the surrounding components weighted according to their hierarchical distance. In the local mean-field limit N→∞, block averages of diffusions within a hierarchical distance k, on an appropriate time scale, are expected to perform a diffusion with local diffusion rate F ^(k) g, where $F^{(k)} g = (F_{c_k } \circ ... \circ F_{c_1 } )$ g is the kth iterate of renormalization transformations F _c (c>0) applied to g. Here the c _k measure the strength of the interaction at hierarchical distance k. We identify F _c and study its orbit (F ^(k) g) _k≥0. We show that there exists a “fixed shape” g* such that lim _k→∞ σ_k F ^(k) g = g* for all g, where the σ _k are normalizing constants. In terms of the infinite system, this property means that there is complete universal behavior on large space-time scales. Our results extend earlier work for d = 1 and $\bar D = [0,1]$, resp. [0, ∞). The renormalization transformation F _c is defined in terms of the ergodic measure of a d-dimensional diffusion. In d = 1 this diffusion allows a Yamada–Watanabe-type coupling, its ergodic measure is reversible, and the renormalization transformation F _c is given by an explicit formula. All this breaks down in d≥2, which complicates the analysis considerably and forces us to new methods. Part of our results depend on a certain martingale problem being well-posed.