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Source author record

Marco Romito

Marco Romito appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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math.PRmath.APmath-phmath.MPMachine Learningmath.CASound

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Published work

18 published item(s)

preprint2026arXiv

Stochastic Scaling Limits and Synchronization by Noise in Deep Transformer Models

We prove pathwise convergence of the layerwise evolution of tokens in a finite-depth, finite-width transformer model with MultiLayer Perceptron (MLP) blocks to a continuous-time stochastic interacting particle system. We also identify the stochastic partial differential equation describing the evolution of the tokens' distribution in this limit and prove propagation of chaos when the number of such tokens is large. The bounds we establish are quantitative and the limits we consider commute. We further prove that the limiting stochastic model displays synchronization by noise and establish exponential dissipation of the interaction energy on average, provided that the common noise is sufficiently coercive relative to the deterministic self-attention drift. We finally characterize the activation functions satisfying the former condition.

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preprint2026arXiv

Universality in Deep Neural Networks: An approach via the Lindeberg exchange principle

We consider the infinite-width limit of a fully connected deep neural network with general weights, and we prove quantitative general bounds on the $2$-Wasserstein distance between the network and its infinite-width Gaussian limit, under appropriate regularity assumptions on the activation function. Our main tool is a Lindeberg principle for Deep Neural Networks, which we use to successively replace the weights on each layer by Gaussian random variables.

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preprint2022arXiv

Yet another notion of irregularity through small ball estimates

We introduce a new notion of irregularity of paths, in terms of control of growth of the size of small balls by means of the occupation measure of the path. This notion ensures Besov regularity of the occupation measure and thus extends the analysis of Catellier and Gubinelli (2016) to general Besov spaces. On stochastic processes this notion is granted by suitable properties of local non-determinism.

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preprint2020arXiv

A Central Limit Theorem for Gibbsian Invariant Measures of 2D Euler Equations

We consider Canonical Gibbsian ensembles of Euler point vortices on the 2-dimensional torus or in a bounded domain of R 2 . We prove that under the Central Limit scaling of vortices intensities, and provided that the system has zero global space average in the bounded domain case (neutrality condition), the ensemble converges to the so-called Energy-Enstrophy Gaussian random distributions. This can be interpreted as describing Gaussian fluctuations around the mean field limit of vortices ensembles. The main argument consists in proving convergence of partition functions of vortices and Gaussian distributions.

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preprint2020arXiv

Decay of Correlation Rate in the Mean Field Limit of Point Vortices Ensembles

We consider the Mean Field limit of Gibbsian ensembles of 2-dimensional point vortices on the torus. It is a classical result that in such limit correlations functions converge to 1, that is, point vortices decorrelate: we compute the rate at which this convergence takes place by means of Gaussian integration techniques, inspired by the correspondence between the 2-dimensional Coulomb gas and the Sine-Gordon Euclidean field theory.

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preprint2015arXiv

Hölder regularity of the densities for the Navier--Stokes equations with noise

We prove that the densities of the finite dimensional projections of weak solutions of the Navier-Stokes equations driven by Gaussian noise are bounded and Hölder continuous, thus improving the results of Debussche and Romito [DebRom2014]. The proof is based on analytical estimates on a conditioned Fokker-Planck equation solved by the density, that has a "non-local" term that takes into account the influence of the rest of the infinite dimensional dynamics over the finite subspace under observation.

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preprint2014arXiv

Global regularity for a logarithmically supercritical hyperdissipative dyadic equation

We prove global existence of smooth solutions for a slightly supercritical dyadic model. We consider a generalized version of the dyadic model introduced by Katz-Pavlovic [2005] and add a viscosity term with critical exponent and a supercritical correction. This model catches for the dyadic a conjecture that for Navier-Stokes equations was formulated by Tao [2009].

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preprint2014arXiv

Time regularity of the densities for the Navier--Stokes equations with noise

We prove that the density of the law of any finite dimensional projection of solutions of the Navier--Stokes equations with noise in dimension $3$ is Hölder continuous in time with values in the natural space $L^1$. When considered with values in Besov spaces, Hölder continuity still holds. The Hölder exponents correspond, up to arbitrarily small corrections, to the expected diffusive scaling.

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preprint2013arXiv

Local existence and uniqueness for a two-dimensional surface growth equation with space--time white noise

We study local existence and uniqueness for a surface growth model with space-time white noise in 2D. Unfortunately, the direct fixed-point argument for mild solutions fails here, as we do not have sufficient regularity for the stochastic forcing. Nevertheless, one can give a rigorous meaning to the stochastic PDE and show uniqueness of solutions in that setting. Using spectral Galerkin method and any other types of regularization of the noise, we obtain always the same solution.

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preprint2012arXiv

Existence of densities for the 3D Navier-Stokes equations driven by Gaussian noise

We prove three results on the existence of densities for the laws of finite dimensional functionals of the solutions of the stochastic Navier-Stokes equations in dimension 3. In particular, under very mild assumptions on the noise, we prove that finite dimensional projections of the solutions have densities with respect to the Lebesgue measure which have some smoothness when measured in a Besov space. This is proved thanks to a new argument inspired by an idea introduced in Fournier and Printems (2010).

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preprint2011arXiv

Rényi Information Measures for Spectral Change Detection

Change detection within an audio stream is an important task in several domains, such as classification and segmentation of a sound or of a music piece, as well as indexing of broadcast news or surveillance applications. In this paper we propose two novel methods for spectral change detection without any assumption about the input sound: they are both based on the evaluation of information measures applied to a time- frequency representation of the signal, and in particular to the spectrogram. The class of measures we consider, the Rényi entropies, are obtained by extending the Shannon entropy definition: a biasing of the spectrogram coefficients is realized through the dependence of such measures on a parameter, which allows refined results compared to those obtained with standard divergences. These methods provide a low computational cost and are well-suited as a support for higher level analysis, segmentation and classification algorithms.

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preprint2011arXiv

Uniqueness and blow-up for the noisy viscous dyadic model

We consider the dyadic model with viscosity and additive Gaussian noise as a simplified version of the stochastic Navier-Stokes equations, with the purpose of studying uniqueness and emergence of singularities. We prove path-wise uniqueness and absence of blow-up in the intermediate intensity of the non-linearity, morally corresponding to the 3D case, and blow-up for stronger intensity. Moreover, blow-up happens with probability one for regular initial data.

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preprint2010arXiv

Local existence and uniqueness in the largest critical space for a surface growth model

We show the existence and uniqueness of solutions (either local or global for small data) for an equation arising in different aspects of surface growth. Following the work of Koch and Tataru we consider spaces critical with respect to scaling and we prove our results in the largest possible critical space such that weak solutions are defined. The uniqueness of global weak solutions remains unfortunately open, unless the initial conditions are sufficiently small.

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preprint2010arXiv

On Leray's problem for almost periodic flows

We prove existence and uniqueness for fully-developed (Poiseuille-type) flows in semi-infinite cylinders, in the setting of (time) almost-periodic functions. In the case of Stepanov almost-periodic functions the proof is based on a detailed variational analysis of a linear "inverse" problem, while in the Besicovitch setting the proof follows by a precise analysis in wave-numbers. Next, we use our results to construct a unique almost periodic solution to the so called "Leray's problem" concerning 3D fluid motion in two semi-infinite cylinders connected by a bounded reservoir. In the case of Stepanov functions we need a natural restriction on the size of the flux, while for Besicovitch solutions certain limitations on the generalized Fourier coefficients are requested.

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