Renormalization of Hierarchically Interacting Isotropic Diffusions

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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 JOURNAL/jstp/04.02/00010029-199809310-00010/OV0430/v/2017-10-11T045013Z/r/image-png ⊂ ℝd (d ≥ 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: JOURNAL/jstp/04.02/00010029-199809310-00010/OV0430/v/2017-10-11T045013Z/r/image-png →[0,∞) 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 = (Fck ˆ … ˆ Fc1) g is the kth iterate of renormalization transformations Fc (c > 0) applied to g. Here the ck measure the strength of the interaction at hierarchical distance k. We identify Fc and study its orbit (F(k)g)k ≥ 0. We show that there exists a “fixed shape” g * such that limk → ∞ σkF(k)g = g * for all g, where the σk 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 JOURNAL/jstp/04.02/00010029-199809310-00010/OV0430/v/2017-10-11T045013Z/r/image-png = [0, 1], resp. [0, ∞). The renormalization transformation Fc 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 Fc 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.

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