Kolloquium Holzegel/Seis/Weber

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Anke Pietsch

Samuel Punshon-Smith (Tulane University, New Orleans): Annealed mixing for advection by stochastic velocity fields

Tuesday, 18.06.2024 14:15 im Raum SRZ 205

Mathematik und Informatik

We consider the long-time behavior of a passive scalar advected by an incompressible velocity field. When the flow is uniformly hyperbolic, it is well known that it is possible to construct special anisotropic Sobolev spaces where the solution operator becomes quasi-compact with a spectral gap, yielding exponential decay in these spaces. In the non-autonomous and non-uniformly hyperbolic case this approach breaks down. In this talk, I will discuss how in the setting of stochastic velocity fields one can recover certain averaged decay estimates using pseudo differential operators to obtain exponential decay of solutions to the transport equation from H^{-\delta} to H^{-\delta} -- a property we call annealed mixing. As a result, we show that (under certain conditions on the velocity) the Markov process obtained by the advection equation with a random source term has a unique stationary measure describing the statistics of "ideal" scalar turbulence.



Angelegt am 08.04.2024 von Anke Pietsch
Geändert am 10.06.2024 von Anke Pietsch
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Kolloquium Holzegel/Seis/Weber
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Anke Pietsch

Francis Nier (Université Sorbonne Paris Nord): Persistent homology and small eigenvalues of Witten and Bismut's hypoelliptic Laplacian

Tuesday, 02.07.2024 14:15 im Raum SRZ 205

Mathematik und Informatik

After the two historical descriptions by Einstein and Langevin of Brownian motion, the now well known generators acting on p-forms, are on one side the Witten Laplacian (Einstein) and on the other side Bismut's hypoelliptic Laplacian (Langevin). The accurate computation of exponentially small eigenvalues has many applications, in particular for the design of effective molecular dynamics algorithms. In the case of the Witten Laplacian, I will present the result obtained a few years ago with D. Le Peutrec and C. Viterbo, which makes the connection between the various exponential scales of small eigenvalues and the bar code of persistent homology. This provides a natural topological extension of the well known Arrhenius law in the scalar case, for general potential functions not assumed to be Morse. I will also present the more recent result obtained with X. Sang and F. White, which provides the same determination of the different spectral exponential scales in terms of the persistent homology bar code, in the double asymptotic regime of large friction and small temperature for Bismut's hypoelliptic Laplacian.



Angelegt am 08.04.2024 von Anke Pietsch
Geändert am 08.04.2024 von Anke Pietsch
[Edit | Vorlage]

Kolloquium Holzegel/Seis/Weber