July 2021: Professor Lucia Caramellino is visiting the Bézout LabEx and is hosted by Aurélien Alfonsi (CERMICS) and Vlad Bally (LAMA) for two weeks.

Lucia Caramellino is a researcher at the university of Roma Tor Vergata, department Mathematics.

They work on the regularity for the semigroup of jump equations. More precisely, following the results stated in [1], they study the regularity (i.e. existence and regularity of the density) of the Markov semigroup associated with a jump equation. In [1], general results on the regularity of Markov semigroups are studied, based on [2]. The current work concerns jump equations driven by a homogeneous Poisson point measure. In order to use the results in [1], the “small jumps” are replaced by a diffusive term, a technique already used in the literature but in a different context (mainly for simulation problems). This approach gives rise to a sequence of regular Markov semigroups converging, in a suitable way, to the Markov semigroup associated with the jump equation at hand. But the sequence of the densities blows up, so the “interpolation technique” from [1] is then used to recover the limit density and its regularity properties.

This visit will be followed by another 4 weeks visit that is planned in September 2021. A possible topic of research will be the analysis of the high order approximation schemes developed in [3] for the total variation and for jump processes by using the method developed in [4]. Another one is a development of [5] to include infinitely many “small jumps”.

References. [1] V. Bally, L. Caramellino (2021+). Transfer of regularity for Markov semigroups by using an interpolation technique. Journal of Stochastic Analysis, to appear. [2] V. Bally, L. Caramellino (2017). Convergence and regularity of probability laws by using an interpolation method. Annals of Probability.

[3] A. Alfonsi, V. Bally (2021+). A generic construction for high order approximation schemes of semigroups using random grids. Numerische Mathematik, to appear.

[4] V. Bally, L. Caramellino, G. Poly (2020). Regularization lemmas and convergence in total variation. Electronic Journal of Probability.

[5] A. Alfonsi, V. Bally (2021) Construction of Boltzmann and Mac Kean Vlasov type flows (the sewing lemma approach), Arxiv preprint.