Multiscale homogenization with bounded ratios and anomalous slow diffusion. (English) Zbl 1205.76223
Summary: We show that the effective diffusivity matrix \(D(V^n)\) for the heat operator \(\partial_t-(\frac12\Delta-\nabla V^n\nabla)\) in a periodic potential \(V_n=\sum_{k=0}^nU_k(x/R_k)\) obtained as a superposition of Hölder-continuous periodic potentials \(U_k\) (of period \({\mathbb T}^d:={\mathbb R}^d/{\mathbb Z}^d,\;d\in{\mathbb N}^*, U_k(0)=0\)) decays exponentially fast with the number of scales when the scale ratios \(R_{k+1}/R_k\) are bounded above and below. From this we deduce the anomalous slow behavior for a Brownian motion in a potential obtained as a superposition of an infinite number of scales, \(dy_t=d\omega_t -\nabla V^\infty(y_t)dt\).
MSC:
| 76M99 | Basic methods in fluid mechanics |
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
| 60J65 | Brownian motion |
| 76S05 | Flows in porous media; filtration; seepage |
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