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DTSTART;TZID=Europe/Paris:20240402T093000
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DTSTAMP:20260426T194558
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UID:16729-1712050200-1712061000@www.math.ens.psl.eu
SUMMARY:A critical drift-diffusion equation: intermittent behavior
DESCRIPTION:This talk is about a simple but rich model problem at the cross section of stochastic homogenization and singular stochastic PDE: We consider a drift-diffusion process with a time-independent and divergence-free random drift that is of white-noise character. As already realized in the physics literature\, the critical case of two space dimensions is most interesting: The elliptic generator requires a small-scale cut-off for well- posedness\, and one expects marginally super-diffusive behavior on large scales. I will explain the criticality of the two-dimensional case in the introductory course by scaling arguments. \nIn the presence of an (artificial) large-scale cut-off at scale L\, as a consequence of standard stochastic homogenization theory and its notion of a corrector\, there exist harmonic coordinates with a stationary gradient F ; the merit of these coordinates being that under their lens\, the drift-diffusion process turns into a martingale. I will revisit (qualitative) stochastic homogenization theory\nin the introductory course. \nIt has recently been established that the second moments diverge as 𝔼 |F|² ~ √ln L as L ↑ ∞. We show that in this limit\, |F|²/𝔼 |F|² is not equi-integrable\, while |det F |/𝔼 |F|² converges to zero (in probability). This suggests that any limit\, if it exists\, will not admit a simple characterization. \nWe establish this asymptotic behavior by characterizing a proxy F̃  introduced in previous work as the solution of an Itô SDE w. r. t. the variable L\, and which implements the concept of a scale-by-scale homogenization. For this proxy\, we establish 𝔼|F̃ |⁴ ≫ (𝔼|F̃ |²)² and 𝔼(det F̃ )² ≲ 1. In view of the former property\, we assimilate this phenomenon to intermittency. I will try to elucidate the idea of scale-by-scale homogenization in the introductory course. \nThis is joint work with G. Chatzigeorgiou\, P. Morfe\, L. Wang\, and with C. Wagner.
URL:https://www.math.ens.psl.eu/evenement/felix-otto/
LOCATION:ENS – salle W\, 45 rue d'Ulm\, Paris\, 75005\, France
CATEGORIES:Séminaire Analyse non linéaire et EDP
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