# Lorenzo MarinoENSTA ParisTech · Department of Applied Mathematics

Lorenzo Marino

Doctor of Philosophy

## About

13

Publications

515

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26

Citations

Introduction

My research interests lie at the interface between probability and PDEs and their applications to physics or finance, with particular focus on models which involve Lévy-type processes or integro-partial differential operators such as:
- Regularisation by degenerate and/or Levy noise for ODEs
- Regularity theory for evolution equation, Schauder/Sobolev estimates for hypoelliptic equations
- Stochastic homogenisation of integro-differential operators, diffusive limits of linear Boltzmann equations

## Publications

Publications (13)

We consider the limit of solutions of scaled linear kinetic equations with a reflection-transmission-killing condition at the interface. Both the coefficient describing the probability of killing and the scattering kernel degenerate. We prove that the long-time, large-space limit is the unique solution of a version of the fractional in space heat e...

We study here the effects of a time-dependent second order perturbation to a degenerate Ornstein-Uhlenbeck type operator whose diffusive part can be either local or non-local. More precisely, we establish that some estimates, such as the Schauder and Sobolev ones, already known for the non-perturbed operator still hold, and with the same constants,...

We consider the limit of solutions of scaled linear kinetic equations with a reflection-transmission-absorption condition at the interface. Both the coefficient describing the probability of absorption and the scattering kernel degenerate. We prove that the long-time, large-space limit is the unique solution of a version of the fractional in space...

After a general introduction about the regularization by noise phenomenon in the degenerate setting, the first part of this PhD thesis focuses at establishing the Schauder estimates, a useful analytical tool to prove also the well-posedness of stochastic differential equations (SDEs), for two different classes of Kolmogorov equations under a weak H...

We consider a possibly degenerate Kolmogorov-Ornstein-Uhlenbeck operator of the form L = Tr(BD 2) + Az, D , where A, B are N x N matrices, z $\in$ R N , N $\ge$ 1, which satisfy the Kalman condition which is equivalent to the hypoellipticity condition. We prove the following stability result: the Schauder and Sobolev estimates associated with the c...

In this article, we study the effects of the propagation of a non-degenerate L{\'e}vy noise through a chain of deterministic differential equations whose coefficients are H{\"o}lder continuous and satisfy a weak H{\"o}rmander-like condition. In particular, we assume some non-degeneracy with respect to the components which transmit the noise. Moreov...

We establish global Schauder estimates for integro-partial differential equations (IPDE) driven by a possibly degenerate Lévy Ornstein-Uhlenbeck operator, both in the elliptic and parabolic setting, using some suitable anisotropic Hölder spaces. The class of operators we consider is composed by a linear drift plus a Lévy operator that is comparable...

We establish global Schauder estimates for integro-partial differential equations (IPDE) driven by a possibly degenerate L\'evy Ornstein-Uhlenbeck operator, both in the elliptic and parabolic setting, using some suitable anisotropic H\"older spaces. The class of operators we consider is composed by a linear drift plus a L\'evy operator that is comp...

We provide here global Schauder-type estimates for a chain of integro-partial differential equations (IPDE) driven by a degenerate stable Ornstein-Uhlenbeck operator possibly perturbed by a deterministic drift, when the coefficients lie in some suitable anisotropic Hölder spaces. Our approach mainly relies on a perturbative method based on forward...

We provide here global Schauder-type estimates for a chain of integro-partial differential equations (IPDE) driven by a degenerate stable Ornstein-Uhlenbeck operator possibly perturbed by a deterministic drift, when the coefficients lie in some suitable anisotropic H{\"o}lder spaces. Our approach mainly relies on a perturbative method based on forw...