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Mario Maurelli

Publications and source records attributed to Mario Maurelli.

At least 19 recordsLinked to original sources
arXiv2025

Edwards-Wilkinson limit for a stochastic advection-diffusion PDE

We consider a diffusion in a Gaussian random environment that is white in time and study the large-scale behavior of the quenched density with respect to the Lebesgue measure. We show that under diffusive rescaling, the fluctuations of the density converge to a Gaussian limit, described by an additive stochastic heat equation. In the case where the environment is divergence-free, our result can be interpreted as computing the scaling limit of the first-order correction to the quenched Central Limit Theorem.

math.PR↗
arXiv2024

Anomalous Regularization in Kazantsev-Kraichnan Model

This work investigates a passive vector field which is transported and stretched by a divergence-free Gaussian velocity field, delta-correlated in time and poorly correlated in space (spatially nonsmooth). Although the advection of a scalar field (Kraichnan's passive scalar model) is known to enjoy regularizing properties, the potentially competing stretching term in vector advection may induce singularity formation. We establish that the regularization effect is actually retained in certain regimes. While this is true in any dimension $d\ge 3$, it notably implies a regularization result for linearized 3D Euler equations with stochastic modeling of turbulent velocities, and for the induction equation in magnetohydrodynamic turbulence.

math.PR↗
arXiv2024

Anomalous Regularization in Kraichnan's Passive Scalar Model

We consider the advection of a passive scalar by a divergence free random Gaussian field, white in time and H\"older regular in space (rough Kraichnan's model), a well established synthetic model of passive scalar turbulence. By studying the evolution of negative Sobolev norms, we show an anomalous regularization effect induced by the dynamics: distributional initial conditions immediately become functions of positive Sobolev regularity.

math.PR↗
arXiv2024

Global well posedness and ergodic results in regular Sobolev spaces for the nonlinear Schr\"odinger equation with multiplicative noise and arbitrary power of the nonlinearity

We consider the nonlinear Schr\"odinger equation on the $d$-dimensional torus $\mathbb T^d$, with the nonlinearity of polynomial type $|u|^{2\sigma}u$. For any $\sigma \in \mathbb N$ and $s>\frac d2$ we prove that adding to this equation a suitable stochastic forcing term there exists a unique global solution for any initial data in $H^s(\mathbb T^d)$. The effect of the noise is to prevent blow-up in finite time, differently from the deterministic setting. Moreover we prove existence of invariant measures and their uniqueness under more restrictive assumptions on the noise term.

math.PR↗
arXiv2024

Regularization by rough Kraichnan noise for the generalised SQG equations

We consider the generalised Surface Quasi-Geostrophic (gSQG) equations in $\mathbb R^2$ with parameter $\beta\in (0,1)$, an active scalar model interpolating between SQG ($\beta=1$) and the 2D Euler equations ($\beta=0$) in vorticity form. Existence of weak $(L^1\cap L^p)$-valued solutions in the deterministic setting is known, but their uniqueness is open. We show that the addition of a rough Stratonovich transport noise of Kraichnan type regularizes the PDE, providing strong existence and pathwise uniqueness of solutions for initial data $\theta_0\in L^1\cap L^p$, for suitable values $p\in[2,\infty]$ related to the regularity degree $\alpha$ of the noise and the singularity degree $\beta$ of the velocity field; in particular, we can cover any $\beta\in (0,1)$ for suitable $\alpha$ and $p$ and we can reach a suitable ("critical") threshold. The result also holds in the presence of external forcing $f\in L^1_t (L^1\cap L^p)$ and solutions are shown to depend continuously on the data of the problem; furthermore, they are well approximated by vanishing viscosity and regular approximations. With similar techniques, we also show well-posedness for two-dimensional linear transport equation with random drift, with the same noise.

math.PR↗
arXiv2023

The Ebin-Marsden toolbox for stochastic PDEs: stochastic Euler equations

The Ebin-Marsden theory is a powerful geometric framework for many PDEs from fluid dynamics. In this paper we provide a toolbox to apply the Ebin-Marsden approach to stochastic PDEs, combining tools from infinite-dimensional geometry and stochastic analysis. We showcase our approach in the context of incompressible Euler equation for an ideal fluid with additive noise. Among our main results there are: (i) local well-posedness of maximal solutions by using the Ebin-Marsden framework; (ii) a stochastic version of the celebrated no-loss-no-gain theorem.

math.PR↗
arXiv2023

Well-posedness of a reaction-diffusion model with stochastic dynamical boundary conditions

We study the well-posedness of a nonlinear reaction diffusion partial differential equation system on the half-line coupled with a stochastic dynamical boundary condition, a random system arising from the description of the chemical reaction of sulphur dioxide with calcium carbonate stones. The boundary condition is given by a Jacobi process, solution to a stochastic differential equation with a mean-reverting drift and a bounded diffusion coefficient. The main result is the global existence and the pathwise uniqueness of mild solutions. The proof relies on a splitting strategy, which allows to deal with the low regularity of the dynamical boundary condition.

math.PR↗
arXiv2023

Existence and uniqueness by Kraichnan noise for 2D Euler equations with unbounded vorticity

We consider the 2D Euler equations on $\R^2$ in vorticity form, with unbounded initial vorticity, perturbed by a suitable non-smooth Kraichnan transport noise, with regularity index $\alpha\in (0,1)$. We show weak existence for every $\dot{H}^{-1}$ initial vorticity. Thanks to the noise, the solutions that we construct are limits in law of a regularized stochastic Euler equation and enjoy an additional $L^2([0,T];H^{-\alpha})$ regularity. For every $p>3/2$ and for certain regularity indices $\alpha \in (0,1/2)$ of the Kraichnan noise, we show also pathwise uniqueness for every $L^p$ initial vorticity. This result is not known without noise.

math.PR↗
arXiv2023

No blow-up by nonlinear It\^o noise for the Euler equations

By employing a suitable multiplicative It\^o noise with radial structure and with more than linear growth, we show the existence of a unique, global-in-time, strong solution for the stochastic Euler equations in two and three dimensions. More generally, we consider a class of stochastic partial differential equations (SPDEs) with a superlinear growth drift and suitable nonlinear, multiplicative It\^o noise, with the stochastic Euler equations as a special case within this class. We prove that the addition of such a noise effectively prevents blow-ups in the solution of these SPDEs.

math.PR↗
arXiv2022

Uniform Approximation of 2D Navier-Stokes Equations with Vorticity Creation by Stochastic Interacting Particle Systems

We consider a stochastic interacting particle system in a bounded domain with reflecting boundary, including creation of new particles on the boundary prescribed by a given source term. We show that such particle system approximates 2d Navier-Stokes equations in vorticity form and impermeable boundary, the creation of particles modeling vorticity creation at the boundary. Kernel smoothing, more specifically smoothing by means of the Neumann heat semigroup on the space domain, allows to establish uniform convergence of regularized empirical measures to (weak solutions of) Navier-Stokes equations.

math.AP↗
arXiv2021

A McKean-Vlasov SDE and particle system with interaction from reflecting boundaries

We consider a one-dimensional McKean-Vlasov SDE on a domain and the associated mean-field interacting particle system. The peculiarity of this system is the combination of the interaction, which keeps the average position prescribed, and the reflection at the boundaries; these two factors make the effect of reflection non local. We show pathwise well-posedness for the McKean-Vlasov SDE and convergence for the particle system in the limit of large particle number.

math.PR↗
arXiv2020

Non-explosion by Stratonovich noise for ODEs

We show that the addition of a suitable Stratonovich noise prevents the explosion for ODEs with drifts of super-linear growth, in dimension $d\ge 2$. We also show the existence of an invariant measure and the geometric ergodicity for the corresponding SDE.

math.PR↗
arXiv2020

Large deviations for singularly interacting diffusions

In this paper we prove a large deviation principle (LDP) for the empirical measure of a general system of mean-field interacting diffusions with singular drift (as the number of particles tends to infinity) and show convergence to the associated McKean-Vlasov equation. Along the way, we prove an extended version of the Varadhan Integral Lemma for a discontinuous change of measure and subsequently an LDP for Gibbs and Gibbs-like measures with singular potentials.

math.PR↗
arXiv2020

Regularized vortex approximation for 2D Euler equations with transport noise

We study a mean field approximation for the 2D Euler vorticity equation driven by a transport noise. We prove that the Euler equations can be approximated by interacting point vortices driven by a regularized Biot-Savart kernel and the same common noise. The approximation happens by sending the number of particles $N$ to infinity and the regularization $ε$ in the Biot-Savart kernel to $0$, as a suitable function of $N$.

math.PR↗
arXiv2019

Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness

In this paper linear stochastic transport and continuity equations with drift in critical $L^{p}$ spaces are considered. In this situation noise prevents shocks for the transport equation and singularities in the density for the continuity equation, starting from smooth initial conditions. Specifically, we first prove a result of Sobolev regularity of solutions, which is false for the corresponding deterministic equation. The technique needed to reach the critical case is new and based on parabolic equations satisfied by moments of first derivatives of the solution, opposite to previous works based on stochastic flows. The approach extends to higher order derivatives under more regularity of the drift term. By a duality approach, these regularity results are then applied to prove uniqueness of weak solutions to linear stochastic continuity and transport equations and certain well-posedness results for the associated stochastic differential equation (sDE) (roughly speaking, existence and uniqueness of flows and their $C^α$ regularity, strong uniqueness for the sDE when the initial datum has diffuse law). Finally, we show two types of examples: on the one hand, we present well-posed sDEs, when the corresponding ODEs are ill-posed, and on the other hand, we give a counterexample in the supercritical case.

math.PR↗
arXiv2019

Pathwise McKean-Vlasov Theory with Additive Noise

Abstract. We take a pathwise approach to classical McKean-Vlasov stochastic differential equations with additive noise, as e.g. exposed in Sznitmann [38]. Our study was prompted by some concrete problems in battery modelling [23], and also by recent progrss on rough-pathwise McKean-Vlasov theory, notably Cass-Lyons [10], and then Bailleul, Catellier and Delarue [4]. Such a "pathwise McKean-Vlasov theory" can be traced back to Tanaka [40]. This paper can be seen as an attempt to advertize the ideas, power and simplicity of the pathwise appproach, not so easily extracted from [4, 10, 40], together with a number of novel applications. These include mean field convergence without a priori independence and exchangeability assumption; common noise, càdlàg noise, and reflecting boundaries. Last not least, we generalize Dawson-Gärtner large deviations and the central limit theorem to a non-Brownian noise setting.

math.PR↗
arXiv2019

Incompressible Euler equations with stochastic forcing: a geometric approach

We consider stochastic versions of Euler--Arnold equations using the infinite-dimensional geometric approach as pioneered by Ebin and Marsden. For the Euler equation on a compact manifold (possibly with smooth boundary) we establish local existence and uniqueness of a strong solution (in the stochastic sense) in spaces of Sobolev mappings (of high enough regularity). Our approach combines techniques from stochastic analysis and infinite-dimensional geometry and provides a novel toolbox to establish local well-posedness of stochastic Euler--Arnold equations.

math.PR↗