arXiv subjects

Giacomo Lucertini

Publications and source records attributed to Giacomo Lucertini.

5 recordsLinked to original sources

arXiv2025

Strong regularization by noise for a class of kinetic SDEs driven by symmetric α-stable processes

We establish strong well-posedness for a class of degenerate SDEs of kinetic type with autonomous diffusion driven by a symmetric $α$-stable process under Hölder regularity conditions for the drift term. We partially recover the thresholds for the Hölder regularity that are optimal for weak uniqueness. In general dimension, we only consider $α= 2$ and need an additional integrability assumption for the gradient of the drift: this condition is satisfied by Peano-type functions. In the one-dimensional case we do not need any additional assumption. In the multi-dimensional case, the proof is based on a first-order Zvonkin transform/PDE, while for the one-dimensional case we use a second-order Zvonkin/PDE transform together with a Watanabe-Yamada technique.

math.PR↗

arXiv2024

A kinetic Nash inequality and precise boundary behavior of the kinetic Fokker-Planck equation

In this paper, we prove a kinetic Nash type inequality and adapt it to a new functional inequality for functions in a kinetic Sobolev space with absorbing boundary conditions on the half-space. As an application, we address the boundary behavior of the kinetic Fokker-Planck equations in the half-space. Our main result is the sharp regularity of the solution at the absorbing boundary and grazing set.

math.AP↗

arXiv2024

Optimal Schauder estimates for kinetic Kolmogorov equations with time measurable coefficients

We prove global Schauder estimates for kinetic Kolmogorov equations with coefficients that are Hölder continuous in the spatial variables but only measurable in time. Compared to other available results in the literature, our estimates are optimal in the sense that the inherent Hölder spaces are the strongest possible under the given assumptions: in particular, under a parabolic Hörmander condition, we introduce Hölder norms defined in terms of the intrinsic geometry that the operator induces on the space-time variables. The technique is based on the existence and the regularity estimates of the fundamental solution of the equation. These results are essential for studying backward Kolmogorov equations associated with kinetic-type diffusions, e.g. stochastic Langevin equation.

math.AP↗

arXiv2023

Optimal regularity for degenerate Kolmogorov equations with rough coefficients

We consider a class of degenerate equations satisfying a parabolic Hörmander condition, with coefficients that are measurable in time and Hölder continuous in the space variables. By utilizing a generalized notion of strong solution, we establish the existence of a fundamental solution and its optimal Hölder regularity, as well as Gaussian estimates. These results are key to study the backward Kolmogorov equations associated to a class of Langevin-type diffusions.

math.AP↗

arXiv2022

Strong regularization by noise for kinetic SDEs

In this paper we prove strong well-posedness for a system of stochastic differential equations driven by a degenerate diffusion satisfying a weak-type Hörmander condition, assuming Hölder regularity assumptions on the drift coefficient. This framework encompasses, as particular cases, stochastic Langevin systems of kinetic SDEs. The drift coefficient of the velocity component is allowed to be $α$-Hölder continuous without any restriction on the index $α$, which can be any positive number in $]0,1[$. As the deterministic counterparts of these differential systems are not well-posed, this result can be viewed as a phenomenon known as regularization by noise.

math.PR↗