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Anomalous slow diffusion from perpetual homogenization. (English) Zbl 1042.60049
Summary: This paper is concerned with the asymptotic behavior of solutions of stochastic differential equations $$dy_t=d\omega_t-\nabla V(y_t)\, dt$$, $$y_0=0$$. When $$d=1$$ and V is not periodic but obtained as a superposition of an infinite number of periodic potentials with geometrically increasing periods [$$V(x) = \sum_{k=0}^\infty U_k(x/R_k)$$, where $$U_k$$ are smooth functions of period 1, $$U_k(0)=0$$, and $$R_k$$ grows exponentially fast with $$k$$], we can show that $$y_t$$ has an anomalous slow behavior and we obtain quantitative estimates on the anomaly using and developing the tools of homogenization. Pointwise estimates are based on a new analytical inequality for subharmonic functions. When $$d\geq 1$$ and V is periodic, quantitative estimates are obtained on the heat kernel of $$y_t$$, showing the rate at which homogenization takes place. The latter result proves Davies’ conjecture and is based on a quantitative estimate for the Laplace transform of martingales that can be used to obtain similar results for periodic elliptic generators.

##### MSC:
 60J60 Diffusion processes 34E13 Multiple scale methods for ordinary differential equations 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 60F05 Central limit and other weak theorems 60G44 Martingales with continuous parameter
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