Lucio Galeati

Lucio Galeati
École Polytechnique Fédérale de Lausanne | EPFL · Mathematics Section

Doctor of Philosophy

About

24
Publications
1,633
Reads
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212
Citations
Citations since 2017
24 Research Items
212 Citations
2017201820192020202120222023020406080100120
2017201820192020202120222023020406080100120
2017201820192020202120222023020406080100120
2017201820192020202120222023020406080100120
Introduction
I'm a PostDoc at EPFL, in the group of Prof. Maria Colombo. I work at the intersection of analysis and probability, mostly focusing on regularisation by noise phenomena and (stochastic) fluid dynamics SPDEs.
Additional affiliations
October 2018 - September 2022
University of Bonn
Position
  • PhD
Description
  • PhD under the supervision of Prof. Massimiliano Gubinelli, funded by a BIGS Scholarship.

Publications

Publications (24)
Preprint
Full-text available
We consider multidimensional SDEs with singular drift $b$ and Sobolev diffusion coefficients $\sigma$, satisfying Krylov--R\"ockner type assumptions. We prove several stability estimates, comparing solutions driven by different $(b^i,\sigma^i)$, both for It\^o and Stratonovich SDEs, possibly depending on negative Sobolev norms of the difference $b^...
Preprint
Full-text available
We consider stochastic differential equations (SDEs) driven by a fractional Brownian motion with drift coefficient $b$ that is allowed to be arbitrarily close to criticality in a scaling sense. We develop a comprehensive solution theory that includes strong existence, path-by-path uniqueness, existence of a solution flow of diffeomorphisms, Malliav...
Article
Full-text available
We study distribution dependent stochastic differential equations with irregular, possibly distributional drift, driven by an additive fractional Brownian motion of Hurst parameter $$H\in (0,1)$$ H ∈ ( 0 , 1 ) . We establish strong well-posedness under a variety of assumptions on the drift; these include the choice $$\begin{aligned} B(\cdot ,\mu )=...
Article
We prove the existence of an eddy heat diffusion coefficient coming from an idealized model of turbulent fluid. A difficulty lies in the presence of a boundary, with also turbulent mixing and the eddy diffusion coefficient going to zero at the boundary. Nevertheless, enhanced diffusion takes place. This article is part of the theme issue ‘Scaling t...
Preprint
Full-text available
In this work we consider solutions to stochastic partial differential equations with transport noise, which are known to converge, in a suitable scaling limit, to solution of the corresponding deterministic PDE with an additional viscosity term. Large deviations and Gaussian fluctuations underlying such scaling limit are investigated in two cases o...
Preprint
Full-text available
We study mixing and diffusion properties of passive scalars driven by $generic$ rough shear flows. Genericity is here understood in the sense of prevalence and (ir)regularity is measured in the Besov-Nikolskii scale $B^{\alpha}_{1, \infty}$, $\alpha \in (0, 1)$. We provide upper and lower bounds, showing that in general inviscid mixing in $H^{1/2}$...
Preprint
Full-text available
We study distribution dependent stochastic differential equation driven by a continuous process, without any specification on its law, following the approach initiated in [16]. We provide several criteria for existence and uniqueness of solutions which go beyond the classical globally Lipschitz setting. In particular we show well-posedness of the e...
Preprint
Full-text available
We study distribution dependent stochastic differential equations with irregular, possibly distributional drift, driven by an additive fractional Brownian motion of Hurst parameter $H\in (0,1)$. We establish strong well-posedness under a variety of assumptions on the drift; these include the choice $$B(\cdot,\mu) = f\ast\mu(\cdot) + g(\cdot),\quad...
Preprint
Full-text available
We prove the mixing property for stochastic linear transport equations on the torus, and dissipation enhancement in the viscous case. Our approach works also for the scaling limit of stochastic 2D (inviscid) fluid dynamical equations with transport noise to deterministic viscous equations. Quantitative estimates on the convergence rates are provide...
Preprint
Full-text available
We prove the existence of an eddy heat diffusion coefficient coming from an idealized model of turbulent fluid. A difficulty lies in the presence of a boundary, with also turbulent mixing and the eddy diffusion coefficient going to zero at the boundary. Nevertheless enhanced diffusion takes place.
Article
For some deterministic nonlinear PDEs on the torus whose solutions may blow up in finite time, we show that, under suitable conditions on the nonlinear term, the blow-up is delayed by multiplicative noise of transport type in a certain scaling limit. The main result is applied to the 3D Keller–Segel, 3D Fisher–KPP, and 2D Kuramoto–Sivashinsky equat...
Article
Full-text available
We consider a family of stochastic 2D Euler equations in vorticity form on the torus, with transport-type noises and L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L...
Article
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Nonlinear Young integrals have been first introduced in Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016) and provide a natural generalisation of classical Young ones, but also a versatile tool in the pathwise study of regularisation by noise phenomena. We present here a self-contained account of the theory, focusing on wellposedn...
Article
Full-text available
A stochastic transport linear equation (STLE) with multiplicative space-time dependent noise is studied. It is shown that, under suitable assumptions on the noise, a multiplicative renormalization leads to convergence of the solutions of STLE to the solution of a deterministic parabolic equation. Existence and uniqueness for STLE are also discussed...
Preprint
Full-text available
Nonlinear Young integrals have been first introduced in [Catellier,Gubinelli, SPA 2016] and provide a natural generalisation of classical Young ones, but also a versatile tool in the pathwise study of regularisation by noise phenomena. We present here a self-contained account of the theory, focusing on wellposedness results for abstract nonlinear Y...
Preprint
Full-text available
For some deterministic nonlinear PDEs on the torus whose solutions may blow up in finite time, we show that, under suitable conditions on the nonlinear term, the blow-up is delayed by multiplicative noise of transport type in a certain scaling limit. The main result is applied to the 3D Keller-Segel, 3D Fisher-KPP and 2D Kuramoto-Sivashinsky equati...
Preprint
Full-text available
We investigate the regularizing effect of certain additive continuous perturbations on SDEs with multiplicative fractional Brownian motion (fBm). Traditionally, a Lipschitz requirement on the drift and diffusion coefficients is imposed to ensure existence and uniqueness of the SDE. We show that suitable perturbations restore existence, uniqueness a...
Preprint
Full-text available
We show that generic H\"older continuous functions are $\rho$-irregular. The property of $\rho$-irregularity has been first introduced by Catellier and Gubinelli (Stoc. Proc. Appl. 126, 2016) and plays a key role in the study of well-posedness for some classes of perturbed ODEs and PDEs. Genericity here is understood in the sense of prevalence. As...
Preprint
Full-text available
We analyse the effect of a generic continuous additive perturbation to the well-posedness of ordinary differential equations. Genericity here is understood in the sense of prevalence. This allows us to discuss these problems in a setting where we do not have to commit ourselves to any restrictive assumption on the statistical properties of the pert...
Preprint
Full-text available
We consider a family of stochastic 2D Euler equations in vorticity form on the torus, with transport type noises and $L^2$-initial data. Under a suitable scaling of the noises, we show that the solutions converge weakly to that of the deterministic 2D Navier--Stokes equations. Consequently, we deduce that the weak solutions of the stochastic 2D Eul...
Preprint
Full-text available
A stochastic transport linear equation (STLE) with multiplicative space-time dependent noise is studied. It is shown that, under suitable assumptions on the noise, a multiplicative renormalization leads to convergence of the solutions of STLE to the solution of a deterministic parabolic equation. Existence and uniqueness for STLE are also discussed...

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Projects

Project (1)
Project
Collection of works in which we explore suitable scaling limits and regularization phenomena for linear and nonlinear PDEs perturbed by a Stratonovich transport noise.