Cari tutti,
lunedì 25 marzo presso la Sala Seminari del Dipartimento di Matematica dell'Università degli studi di Pisa si terranno i seguenti seminari:
(Dear all, on Monday, March 25th, the following two talks will take place in Sala Seminari, Dipartimento di Matematica, University of Pisa:)
ORE 14:00 -- Stefano Pagliarani (Università degli studi di Udine) **Contraction methods for a class of McKean-Vlasov (mean-field) SDEs with jumps.**
ORE 15:00 -- Martin Saal (University of Darmstadt) **White noise solutions for (m)SQG**
Tutti gli interessati sono invitati a partecipare.
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Abstract S. Pagliarani:
We consider two prototype classes of McKean-Vlasov (mean-field) SDEs with jumps. In the first case, the coefficient is assumed to be affine in the state-variable, only measurable in the law, and the dynamics allow for Lévy jumps. We study the equivalent functional fixed-point equation for the unknown time-dependent coefficients of the associated Markovian SDE. By proving a contraction property for the functional map in a suitable normed space, we infer existence and uniqueness results for the MK-V SDE, and derive a discretized Picard iteration method that approximates the law of the solution. Numerical illustrations show the effectiveness of the method, which appears to be appropriate to handle multi-dimensional settings. We finally describe possible extensions and generalizations to more general settings. The second class of MKV SDEs that we consider allows for self-exicitng jumps, which amounts to having jumps through hitting the boundary in the equivalent large particle system. The corresponding PDE problem is a particular instance of free-boundary problems known as supercooled Stefan-like problems. We write a Volterra equation for the free boundary and prove contraction results that are useful to determine the properties of the solution and to provide convergent numerical schemes.
Abstract M. Saal:
The inviscid surface quasigeostrophic equation (SQG) describes (roughly speaking) the temperature in a rapidly rotating stratified fluid which is transported by the velocity field. The velocity field is connected to the temperature via Riesz-transform, which are singular integral operators. It has applications in both meteorological and oceanic flows, while in mathematics it is often used as a toy model for the 3D Euler equations due to some structural similarities of these equations. We give a brief overview on versions of the SQG equation and of the known mathematical results. For a modified version (mSQG) with a smoother velocity field, which links the SQG equation to the vorticity formulation of the 2D Euler equations, we will show that by using a special symmetry in the kernel a white noise solution to mSQG can be constructed. Finally, we give some comments on the difficulties of our approach in the case of SQG itself.