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SUMMARY:Superdiffusivity for a diffusion in a critically-correlated incomp
 ressible random drift - Scott Armstrong\, Sorbonne Université
DTSTART:20250512T130000Z
DTEND:20250512T140000Z
UID:TALK231874@talks.cam.ac.uk
CONTACT:Amelie Justine Loher
DESCRIPTION:We consider an advection-diffusion (or "passive scalar") equat
 ion with a divergence-free vector field\, which is a stationary random fie
 ld exhibiting "critical" correlations. Predictions from physicists in the 
 80s state that\, almost surely\, this equation should behave like a heat e
 quation at large scales\, but with a diffusivity that diverges as the squa
 re root of the log of the scale. In joint work with Ahmed Bou-Rabee and Tu
 omo Kuusi\, we give a rigorous proof of this prediction using an iterative
  quantitative homogenization procedure\, which is a way of formalizing a r
 enormalization group argument. The idea is to consider a scale decompositi
 on of the vector field\, and coarse-grain the equation\, scale-by-scale. T
 he random swirls of the vector field at each scale enhance the effective d
 iffusivity. As we zoom out\, we obtain an ODE for the effective diffusivit
 y as a function of the scale\, allow us to deduce that it diverges at the 
 predicted rate. Meanwhile\, new coarse-graining arguments allow us to rigo
 rously integrate out the smaller scales in the equation and prove the resu
 lt.
LOCATION:MR13
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