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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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