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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:A critical drift-diffusion equation: intermittent
behavior - Felix Otto (Max-Planck-Institut für Mat
hematik\, Leipzig)
DTSTART;TZID=Europe/London:20240709T091500
DTEND;TZID=Europe/London:20240709T101500
UID:TALK215566AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/215566
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 dive
rgence-freerandom drift that is of white-noise cha
racter. As already realized in the physics literat
ure\, the critical case of two space dimensions is
most interesting: The elliptic generator requires
a small-scale cut-off for wellposedness\, and on
e expects marginally super-diffusive behavior on l
arge scales.\nIn the presence of an (artificial) l
arge-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 me
rit of these coordinates being that under their le
ns\, the drift-diffusion process turns into a mart
ingale.\nIt has recently been established that the
second moments diverge as E|F| 2 &sim\; &radic\;
ln L as L &uarr\; &infin\;. We show that in this l
imit\, |F| 2/E|F| 2 is not equi-integrable\, while
|detF|/E|F| 2 converges to zero (in probability).
\nWe establish this asymptotic behavior by charact
erizing a proxy F&tilde\; introduced in previous
work as the solution of an It&circ\;o SDE w. r. t.
the variable L\, and which implements the concep
t of a scale-by-scale homogenization. This itself
is close to a tensorial version of a stochastic e
xponential\, with many similarities to the Gaussi
an Multiplicative Chaos. In line with this\, we es
tablish E|F&tilde\;| 4 ≫ (E|F&tilde\;| 2 ) 2 and E
(detF&tilde\;) 2 ≲ 1. Inview of the former propert
y\, we assimilate this phenomenon to intermittenc
y.\nThis is joint work with G. Chatzigeorgiou\, P.
Morfe\, L. Wang\, and withC. Wagner.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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