A critical drift-diffusion equation: intermittent behavior

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• Felix Otto (Max-Planck-Institut für Mathematik, Leipzig)
• Tuesday 09 July 2024, 09:15-10:15
• Seminar Room 1, Newton Institute.

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SSDW01 - Self-interacting processes

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-freerandom 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 wellposedness, and one expects marginally super-diffusive behavior on large scales. In 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. It 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 |detF|/E|F| 2 converges to zero (in probability). We establish this asymptotic behavior by characterizing a proxy F˜ introduced in previous work as the solution of an Itˆo SDE w. r. t. the variable L, and which implements the concept of a scale-by-scale homogenization. This itself is close to a tensorial version of a stochastic exponential, with many similarities to the Gaussian Multiplicative Chaos. In line with this, we establish E|F˜| 4 ≫ (E|F˜| 2 ) 2 and E(detF˜) 2 ≲ 1. Inview of the former property, we assimilate this phenomenon to intermittency. This is joint work with G. Chatzigeorgiou, P. Morfe, L. Wang, and withC. Wagner.

This talk is part of the Isaac Newton Institute Seminar Series series.

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