Large Space-Time Scale Behavior of Linearly Interacting Diffusions
نویسندگان
چکیده
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 a compact convex set D ⊂ 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 : D → [0,∞) chosen from an appropriate class; (2) a linear drift towards 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 ◦ · · · ◦ Fc1)g is the k-th 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 g)k≥0. We show that there exists a ‘fixed shape’ g∗ such that limk→∞ σk F (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 D = [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
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تاریخ انتشار 1999