July 2021, Sep. 2021: Professor Lucia Caramellino is visiting the Bézout LabEx and is hosted by Aurélien Alfonsi (CERMICS) and Vlad Bally (LAMA) for six weeks in total.
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 , they study the regularity (i.e. existence and regularity of the density) of the Markov semigroup associated with a jump equation. In , general results on the regularity of Markov semigroups are studied, based on . The current work concerns jump equations driven by a homogeneous Poisson point measure. In order to use the results in , 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  is then used to recover the limit density and its regularity properties.
A possible topic of research will be the analysis of the high order approximation schemes developed in  for the total variation and for jump processes by using the method developed in . Another one is a development of  to include infinitely many “small jumps”.
References.  V. Bally, L. Caramellino (2021+). Transfer of regularity for Markov semigroups by using an interpolation technique. Journal of Stochastic Analysis, to appear.  V. Bally, L. Caramellino (2017). Convergence and regularity of probability laws by using an interpolation method. Annals of Probability.
 A. Alfonsi, V. Bally (2021+). A generic construction for high order approximation schemes of semigroups using random grids. Numerische Mathematik, to appear.
 V. Bally, L. Caramellino, G. Poly (2020). Regularization lemmas and convergence in total variation. Electronic Journal of Probability.
 A. Alfonsi, V. Bally (2021) Construction of Boltzmann and Mac Kean Vlasov type flows (the sewing lemma approach), Arxiv preprint.