Dear all,

We announce two MAP seminars for Thursday, 28 November, in Aula Riunioni, starting at 11:10. You can also attend the seminar online on Google Meet: https://meet.google.com/vtt-bhkm-tzc

⟶ *Schedule: *

- 11:10 - 12:00 - *Alessandro Pinzi* - Continuity equation on random measures and a new superposition principle for the non-local case*;* - 12:00 - 12:50 - *Fanch Coudreuse *- Second Order Estimates in the JKO scheme for the Fokker-Planck equation.

More information is provided below.

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⟶ *Speaker 1 **:* *Alessandro Pinzi* (Università Bocconi)

⟶ *Title **:* *Continuity equation on random measures and a new superposition principle for the non-local case*

⟶ *Abstract **: *The continuity equation plays an important role in different contexts: it models many physical phenomena, due to the conservation of mass, but it's also fundamental for more theoretical reasons, e.g. understanding the geometry of the Wasserstein space 𝒫p(ℝd). We put the focus on its relation with the classic ODE, thanks to what is known as `superposition principle' in literature. In particular, we investigate the non-local case and we prove a new superposition principle that links three different equations on three different spaces: an abstract continuity equation over the space of random probability measure 𝒫p(𝒫p(ℝd )), the non-local continuity equation over 𝒫p(ℝd) and an interacting particle systems on ℝd. The presentation is based on a joint work with Giuseppe Savaré.

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⟶ *Speaker 2 **:* *Fanch Coudreuse *(École Normale Supérieure de Lyon)

⟶ *Title **:* *Second Order Estimates in the JKO scheme for the Fokker-Planck equation*

⟶ *Abstract **: *Introduced by Jordan, Kinderlehrer, and Otto, the JKO scheme is a time discretization method based on optimal transport theory, designed for several parabolic (and possibly degenerate) equations. It can be interpreted as an implicit Newton-type scheme in the Wasserstein space, where the usual Euclidean distance is replaced by its optimal transport counterpart. While the convergence of the flow under various levels of generality is well-understood in the weak setting, one can hope to recover information that is known to hold for the continuous-time equation in the case of certain specific flows. In this talk, I will present second-order estimates of the Li-Yau and Hamilton matrix forms, which hold for the JKO scheme associated with the Fokker-Planck equation, either on the torus or on the whole space, and I will discuss some applications of these estimates. If time permits, I will also explain how to modify these estimates for the porous medium equation.

This is based on joint work with F. Santambrogio, which will be published soon. ---------------------------------------------------------------

We remind you that these seminars are addressed to everybody: the first half of the seminar is dedicated to the introduction of notions and tools needed in the second half, which is focused on research topics.

See you soon,

The organizers: Michele Caselli Francesco Malizia Jeremy Mirmina Filippo Paiano Leonardo Roveri

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