Renormalization of hierarchically interacting isotropic diffusions

We study a renormalization transformation arising in an infinite system of interacting diffusions. The components of the system are labeled by the N-d imensional hierarchical lattice (N greater than or equal to 2) and take val ues in the closure of a compact convex set (D) over bar subset of R-d (d gr eater than or equal to 1). Each component starts at some BED and is subject to two motions: (Ij an isotropic diffusion according to a local diffusion rate g: (D) over bar [0, infinity) chosen from an appropriate class; (2) a linear drift toward an average of the surrounding components weighted accor ding to their hierarchical distance. In the local mean-held limit N --> inf inity, block averages of diffusions within a hierarchical distance k, on an appropriate time scale, are expected to perform a diffusion with local dif fusion rate F-(k)g, where F-(k)g = (F-ck circle...circle F-c1) g is the kth iterate of renormalization transformations F-c (c > 0) applied to g. Here the Ck measure the strength of the interaction at hierarchical distance k. We identify F, and study its orbit (F-(k)g)(k greater than or equal to 0). We show that there exists a "fixed shape" g* such that lim(k-->infinity) si gma(k)F((k))g = g* for all g, where the sigma(k) are normalizing constants. In terms of the infinite system, this property means that there is complet e universal behavior on large space-time scales. Our results extend earlier work for d=1 and (D) over bar=[0, 1], resp. [0, infinity). The renormaliza tion transformation F, is defined in terms of the ergodic measure of a d-di mensional diffusion. In d=1 this diffusion allows a Yamada-Watanabe-type co upling, its ergodic measure is reversible, and the renormalization transfor mation F, is given by an explicit formula. All this breaks down in d greate r than or equal to 2, which complicates the analysis considerably and force s us to new methods. Part of our results depend on a certain martingale pro blem being well-posed.