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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.

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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