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SUMMARY:Energy solutions and (super-)critical singular S(P)DEs
DTSTART:20260115T160000
DTEND:20260115T164000
DTSTAMP:20260916T092557Z
UID:882b31b6c775116d0d750ce25c4864edeffefb0294b28340e8ecc21b
CATEGORIES:Conferences - Seminars
DESCRIPTION:Lukas Grafner\nIn this talk I will give an overview over recen
 t progress in the understanding of certain singular stochastic dynamics in
  the so-called scaling-critical and -supercritical regime. Our approach ce
 nters around energy solutions as introduced by Jara\, Gonçalves and Gubin
 elli. Leveraging the underlying Markov structure\, we prove weak unique so
 lvability of such solutions for certain SPDEs such as critical stochastic 
 surface quasigeostrophic equations and the critical fractional 1-Dstochast
 ic Burgers equation. The solutions are diffusively scaling-invariant and n
 on-Gaussian as processes. We will further point to several more recent rob
 ustifications of this method regarding the structure of the underlying equ
 ation.\n\nWe also develop a theory of energy solutions in the finite-dimen
 sional setting of SDEs with distributional drift and show weak well-posedn
 ess of such solutions for drifts which lie in certain supercritical Besov 
 spaces with negative regularity and can have even more singular but local 
 blow-ups.\n\nMoreover\, we construct the 1-D self-repelling Brownian polym
 er (SRBP) which is formally the solution to an SDE with path-dependent dis
 tributional drift: the negative gradient of the local time of the solution
 . This is achieved by rigorously establishing a certain transformation of 
 the SDE to an SPDE that can be uniquely solved in the sense of energy solu
 tions. We then give a dynamic characterization of the SRBP and show that i
 t is superdiffusive and “nowhere self-avoiding”.\n\nBased on joint wor
 ks with Harry Giles\, Nicolas Perkowski and Shyam Popat.
LOCATION:Bernoulli Center - GA 3 21 https://plan.epfl.ch/?room==GA%203%202
 1
STATUS:CONFIRMED
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