中国科学院数学与系统科学研究院
数学研究所
数学科学全国重点实验室
偏微分方程研讨班
Speaker: Professor Scott Armstrong (Sorbonne University)
Inviter: 阮国兴
Language: English
Title：Coarse-grained ellipticity and the De Giorgi--Nash--Moser theorem
Time&Venue：2026年3月24日（星期二）15:30-16:30 & ZoomID: 373 227 3489 Passcode: AMSS2026
Abstract:
Title：Anomalous diffusivity and regularity for random incompressible flows
Time&Venue：2026年3月26日（星期四）15:30-16:30
& ZoomID: 373 227 3489 Passcode: AMSS2026
Abstract: We consider the behavior of Brownian motion in a "turbulent" drift, that is, a stationary, incompressible random drift field with slowly decaying correlations. In this setting, one expects the variance of the displacement to grow faster than linearly in time, with an exponent determined by the correlation structure of the drift. This behavior was predicted by physicists in 1990 using perturbative renormalization group heuristics. We formulate the problem as a PDE problem via the associated divergence-form drift–diffusion operator, which has self-similar (or multifractal) coefficients. We apply a scale-by-scale coarse-graining scheme to this operator. At each scale, this produces an effective Laplacian whose diffusivity depends on the scale, together with quantitative control of the approximation error. In other words, we use methods originating in quantitative homogenization theory, but we must iteratively perform infinitely many homogenizations, and the operator never “finishes” homogenizing because of its self-similar structure. This may be viewed as a rigorous version of the perturbative RG heuristics. A crucial role is played by anomalous regularization, that is, regularity estimates for solutions that are independent of the bare molecular diffusivity. The work I will describe is based on our joint paper with A. Bou-Rabee and T. Kuusi available here: https://arxiv.org/abs/2601.22142
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