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Franco Flandoli

Publications and source records attributed to Franco Flandoli.

At least 19 recordsLinked to original sources
arXiv2026

An Effective SPDE Model for a Schr\"odinger Equation with a Fluctuating Magnetic Potential

We consider the Schr\"odinger equation for a charged particle in a randomly fluctuating external magnetic field and establish a singular perturbation limit in which the solution converges to a deterministic Schr\"odinger equation with Gaussian fluctuations. We propose a stochastic Schr\"odinger equation incorporating the behavior of both the deterministic average and the fluctuations and show that it has the same limiting behavior as the original model, while providing a simpler way to study the distribution of relevant observables.

math.PR↗
arXiv2026

Exponential decay of mass for inertial coalescing particles with Hamiltonian noise

We study a system of $N$ inertial particles on a two-dimensional torus $\T^2$, evolving under a second-order stochastic dynamics with position-dependent friction $\lambda$ and noise amplitude $\sigma$, and undergoing coalescence at rate $R_0$ when their distance falls below a threshold $\delta$. In the joint small-mass / small-correlation limit $\mu(\eps)\to 0$, $\mu(\eps)/\eps\to\al\in(0,\infty)$, the empirical measure of the surviving particles converges to a stochastic continuity equation with inertial drift~$g_\al$. Assuming that $\sigma$ is tangent to the level sets of a Hamiltonian $H=h_1(x_1)\,h_2(x_2)$ satisfying mild non-degeneracy and convexity-type conditions, and that $\lambda$ and the amplitude $\rho$ of $\sigma$ along $\xi=\nabla^\perp H$ are aligned with $H$, we prove that the expected total mass decays exponentially in time, with an explicit rate depending on $\al$ and on the values of $\lambda$ and $\rho$ on the separatrix $\{H=0\}$. The proof rests on a cell-by-cell analysis of the sign of $\div\,g_\al$ on the level sets of $H$, showing that the inertial drift pushes trajectories toward the separatrix at a quantitative rate.

math.PR↗
arXiv2026

The inertial Itô drift and its applications to particle collision

The small mass $μ$ limit of an inertial system driven by an Ornstein Uhlenbeck fluid force, with correlation time $ε$ going to zero, leads to a first order system with an additional drift, which we call inertial-Itô-drift, depending on the limit $α$ of the ratio $μ/ε$; the drift being zero when $α=0$, corresponding to the Stratonovich integral in the limit equation, as in the Wong-Zakai theory, when applied directly to the first-order system with Ornstein-Uhlenbeck driver. We discuss the application of this result to particles driven by Stokes force;\ we identify inertial centrifugal effects and the so-called turbophoretic effect, as examples of the inertial-Itô-drift. We also analyze concentration effects and their link with the theory of particle collision in turbulent fluids.

math.PR↗
arXiv2026

Stochastic modeling of Fourier modes in two-dimensional turbulence via filtered white noise

Modeling turbulent flows by a random Fourier decomposition is a classical procedure in order to use simplified models of turbulence in heat transport and other applications. We investigate the Fourier time series of two-dimensional Navier-Stokes equations with friction and damping, forced at intermediate scales, and identify significant statistical structures. In particular, we find the existence of a typical time correlation length, and propose a stochastic model for the Fourier components. Finally, we compute the transport of a passive scalar under advection-diffusion dynamics by means of direct numerical simulation of the stochastic damped Navier-Stokes equation and compare it with analytical predictions of the effective diffusion produced by the stochastic model.

math-ph↗
arXiv2026

Optimal response for stochastic differential equations in $\mathbb{T}^d$ with perturbations on the drift term

We study stochastic differential equations on the $d$-dimensional flat torus $\mathbb{T}^d$ with drift and perturbation coefficients in $L^{\infty}(\mathbb{T}^d;\mathbb{R}^d)$ and additive non-degenerate noise. For the associated transfer operators, we analyse the dependence of the stationary measure and of the expectation of a given observable on small perturbations of the drift. In this framework, we prove a linear response formula for the invariant density and for the expectation of a given observable. We then address an optimal response problem, namely the determination of admissible perturbations that maximise the first-order variation of a prescribed observable. We establish existence of optimal perturbations and, in a Hilbert space framework, prove uniqueness and provide an explicit characterisation of the optimiser. This yields a practical Fourier-based numerical method, which we implement in several numerical examples, including both low and high-dimensional settings.

math.DS↗
arXiv2026

Stochastic transport by Gaussian noise

Diffusion with stochastic transport is investigated here when the random driving process is a very general Gaussian process, including Fractional Brownian motion. The purpose is the comparison with a deterministic PDE, which in certain cases represents the equation for the mean value. From this equation we observe a reduced dissipation property for small times and an enhanced diffusion for large times, with respect to delta correlated noise when regularity is higher than the one of Brownian motion, a fact interpreted qualitatively here as a signature of the modified dissipation observed for 2D turbulent fluids due to the inverse cascade. We give results also for the variance of the solution and for a scaling limit of a two-component noise input.

math.PR↗
arXiv2026

The early stage of the motion along the gradient of a concentrated vortex structure

We give a rigorous mathematical result, supported by numerical simulations, of the aggregation of a concentrated vortex blob with an underlying non-constant vorticity field: the blob moves in the direction of the gradient of the field. It is a unique example of a Lagrangian explanation of aggregation of vortex structures of the same sign in 2D inviscid fluids. The result is also extended to almost vertical vortex filaments in a (possibly thin) three-dimensional domain.

math-ph↗
arXiv2026

A Multiplicative-Noise Mechanism for Variability Amplification under Radiative Forcing in an Arctic Energy-Balance Model

We propose and analyse a mechanism by which $\mathrm{CO}_2$-driven radiative forcing can increase Arctic temperature variability in a stochastic Sellers-type energy-balance model. Starting from a fast-slow formulation in which insolation is modelled by a rapidly mean-reverting Ornstein-Uhlenbeck process while temperature evolves on a slow macroweather timescale, a Wong-Zakai reduction leads to a stochastic energy-balance equation with \emph{multiplicative} noise. After linearising around the stable equilibrium $T^{*,λ}$, we derive an explicit expression for the stationary variance of the temperature anomaly and prove that it increases monotonically with the forcing parameter $λ$ whenever $T^{*,λ}$ lies in the ice-sensitive regime of the co-albedo. We then consider a spatial anomaly model and its finite-difference semi-discretisation, obtaining a finite-dimensional SDE. Under natural stability conditions and nonnegative noise correlations, we establish a component-wise monotone increase of the stationary covariance matrix with respect to $λ$, including its off-diagonal entries. In particular, radiative forcing amplifies not only local variances but also the covariance between temperature anomalies at distinct spatial locations, indicating increased similarity in the variability of the anomaly field across space.

math.PR↗
arXiv2026

Anomalous diffusion properties of stochastic transport by heavy-tailed jump processes

In this work, we investigate the large-scale transport properties of a passive scalar advected by a turbulent fluid, modelled as a superposition of divergence-free vector fields, each weighted by an independent symmetric $α$-stable-like process. Motivated by recent works showing that complex small-scale spatial structures often lead to Brownian dispersion, we study if this principle persists when the driving noise exhibits heavy-tailed jump statistics. Our numerical results show a clear dichotomy linked with the tail behaviour of the noise. When considering standard $α$-stable processes, very large jumps survive the interaction with the spatial complexity and yield anomalous, super-diffusive transport. In contrast, when the $α$-stable noise is either truncated or exponentially tempered, suppressing extremely long jumps, the transport undergoes a transition to a classical diffusive regime.

math-ph↗
arXiv2026

Understanding the householder solar panel consumer: A Markovian model and its societal implications

Household adoption of rooftop photovoltaic (PV) systems is central to the green energy transition, yet diffusion depends on social influence and behavioral biases, as well as payback economics. This study develops a parsimonious Markovian model in which households move sequentially from being unengaged (Carbon) to informed, to planning, and finally to adoption (Green). Transition rates are micro-founded by two mechanisms: (i) social contagion/communication, proxied by the current share of adopters, and (ii) economic profitability, proxied by payback time computed from a Net Present Value framework. Novel to this diffusion setting, bounded rationality is introduced via hyperbolic discounting, creating a procrastination loop that delays adoption even when PV is economically attractive in a long-run perspective. Calibrated on the Italian residential PV diffusion path (2006-2020) and assessed in national and regional applications, the model reproduces observed trajectories and enables forward-looking scenario analysis (2020-2026). Results show that policies yielding similar payback improvements can produce different outcomes once present bias is accounted for and that behaviorally informed intervention are stronger. The findings contribute a micro-to-macro bridge between behavioral economics and technology diffusion modeling and imply that effective policy portfolios (and PV business models) should complement incentives with commitment devices and social-norm peer strategies to accelerate PV uptake and its spillover emissions benefits.

physics.soc-ph↗
arXiv2025

The hard membrane process and transport barriers of turbulent flows

Motivated by the phenomenon of transport barriers in fusion plasma devices, we write a mathematical model of heat dispersion in a turbulent fluid with a transport barrier, properly idealized; in a scaling limit of the turbulence model with separation of scales we get a heat equation with space-dependent diffusion coefficient, poorly diffusing near the barrier; then we investigate the scaling limit when the diffused barrier converges to a sharp separating surface and describe the limit by means of the stochastic process called Brownian motion with hard membrane.

math.PR↗
arXiv2025

Hasegawa-Mima equation in bounded domain with singular density

We investigate the well-posedness of Hasegawa-Mima equation (HME) in bounded domain with Dirichlet boundary condition and singular density under different regularity assumptions on the data. Our approach relies on the coupling of a fourth-order regularization of the HME, the spectral Galerkin method and an appropriate regularization of the singular density. Uniqueness holds in the class of solutions with very mild singularities à la Yudovich or with bounded densities.

math.AP↗
arXiv2025

Structural properties in the diffusion of the solar photovoltaic in Italy: individual people/householder vs firms

This paper develops two mathematical models to understand subjects' behavior in response to the urgency of a change and inputs from governments e.g., (subsides) in the context of the diffusion of the solar photovoltaic in Italy. The first model is a Markovian model of interacting particle systems. The second one, instead, is a Mean Field Game model. In both cases, we derive the scaling limit deterministic dynamics, and we compare the latter to the Italian solar photovoltaic data. We identify periods where the first model describes the behavior of domestic data well and a period where the second model captures a particular feature of data corresponding to companies. The comprehensive analysis, integrated with a philosophical inquiry focusing on the conceptual vocabulary and correlative implications, leads to the formulation of hypotheses about the efficacy of different forms of governmental subsidies.

physics.soc-ph↗
arXiv2025

A scaling limit for Vlasov equations with electrostatic fluctuations

We consider a Vlasov equation for a plasma with a given constant magnetic field, and introduce a white noise perturbation of the electric field in the electrostatic approximation, with a discussion of the motivations of such random perturbation. We prove that diffusion in velocity emerges in a suitable scaling limit of the noise, and also discuss the physical relevance of this result.

math.PR↗
arXiv2025

Diffusion properties of small-scale fractional transport models

Stochastic transport due to a velocity field modeled by the superposition of small-scale divergence free vector fields activated by Fractional Gaussian Noises (FGN) is numerically investigated. We present two non-trivial contributions: the first one is the definition of a model where different space-time structures can be compared on the same ground: this is achieved by imposing the same average kinetic energy to a standard Ornstein-Uhlenbeck approximation, then taking the limit to the idealized white noise structure. The second contribution, based on the previous one, is the discover that a mixing spatial structure with persistent FGN in the Fourier components induces a classical Brownian diffusion of passive particles, with suitable diffusion coefficient; namely, the memory of FGN is lost in the space complexity of the velocity field.

cond-mat.stat-mech↗
arXiv2025

The Stefan problem with mushy region as a scaling limit of stochastic PDE with turbulent transport

This work establishes a scaling limit theorem for the Stefan problem incorporating a mushy region, demonstrating that solutions to stochastic variants with turbulent transport terms converge to the solution to a deterministic partial differential equation. The analysis builds upon recent advances in stochastic phase-change modeling and turbulent flow mathematics in [5]. In the physical interpretation of an ice melting process, our result shows that turbulence accelerates ice melting.

math.AP↗
arXiv2025

An Existence Result for a Stochastic Stefan Problem With Mushy Region and Turbulent Transport Noise

This work is devoted to the proof of the existence of a martingale solution for a complex version of the stochastic Stefan problem. This particular formulation incorporates two important features: a mushy region and turbulent transport within the liquid phase. While our approach bears similarities to porous media equations, it differs in a crucial aspect. Instead of using the typical framework for such equations, we have chosen to work within an L2 space. This choice is motivated by the nature of the operator that characterizes the turbulent noise in our model. The L2 space provides a more natural and appropriate setting for handling this specific operator, allowing us to better capture and analyze the turbulent transport phenomena in the liquid phase of the Stefan problem.

math.AP↗