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Sandra Cerrai

Publications and source records attributed to Sandra Cerrai.

At least 19 recordsLinked to original sources
arXiv2026

Large deviations for long-time occupation measures of stochastic evolution equations with small, asymptotically rough noise

We study the long-time, small-noise behavior of a class of dissipative stochastic evolution equations in a separable Hilbert space, driven by a cylindrical Wiener process whose covariance degenerates to a limiting operator in the strong operator topology. A prototypical example is a stochastic reaction-diffusion equation on a bounded domain with spatially homogeneous but spectrally regularized noise that becomes spatially rough in the limit. We establish a large deviation principle for occupation measures as the time horizon becomes large, the noise intensity tends to zero, and the noise becomes increasingly rough. The result covers a broad class of dissipative equations in infinite dimensions, including those driven by asymptotically rough cylindrical noise whose covariance need not be trace class. Extending the finite-dimensional work of Budhiraja and Zoubouloglou, the infinite-dimensional setting introduces substantial new difficulties: the bound arguments require careful handling of the unbounded evolution and inverse covariance operators, and the increasing roughness of the noise must be balanced against its vanishing amplitude through uniform estimates. Proofs combine analytic semigroup techniques, fractional domain space estimates and a careful treatment of stochastic convolutions in weighted spaces. The rate function is given by a simple explicit formula, the average over the measure of the squared Cameron-Martin cost of canceling the deterministic drift at each point. The proof follows the weak convergence approach based on the Bou\`{e}-Dupuis variational formula and constructs near-optimal controls by alternating travel phases that steer the process between prescribed target states and hold phases that stabilize it near a target while shaping the occupation measure.

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 It\^o drift and its applications to particle collision

The small mass $\mu$ limit of an inertial system driven by an Ornstein Uhlenbeck fluid force, with correlation time $\epsilon$ going to zero, leads to a first order system with an additional drift, which we call inertial-It\^{o}-drift, depending on the limit $\alpha$ of the ratio $\mu/\epsilon$; the drift being zero when $\alpha=0$, 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-It\^{o}-drift. We also analyze concentration effects and their link with the theory of particle collision in turbulent fluids.

math.PR↗
arXiv2026

Stochastic Modified Equations for Stochastic Gradient Descent in Infinite-Dimensional Hilbert Spaces

Inverse problems in scientific computing often require optimization over infinite-dimensional Hilbert spaces. A commonly used solver in such settings is stochastic gradient descent (SGD), where gradients are approximated using randomly sampled sub-objective functions. In this work we study the continuous-time limit of SGD in the small step-size regime. We show that the discrete dynamics can be approximated by a stochastic differential equation (SDE) driven by cylindrical Brownian motion. The analysis extends diffusion-approximation results previously established in Euclidean spaces to the infinite-dimensional setting. Two analytical difficulties arise in this extension. First, the cylindrical nature of the noise requires establishing well-posedness of the resulting stochastic evolution equation through appropriate structural conditions on the covariance operator. Second, since the randomness in SGD originates from discrete sampling while the limiting equation is driven by Gaussian noise, the comparison between the two dynamics must be carried out in a weak sense. We therefore introduce a suitable class of smooth functionals on the Hilbert space and prove that the discrepancy between SGD and the limiting SDE, when evaluated through these functionals, is of second order in the step size. Numerical experiments confirm the predicted convergence behavior.

math.OC↗
arXiv2025

Parabolic scaling of a stochastic wave map with co-normal noise: limit and fluctuations

This paper investigates the parabolic scaling limit of a damped stochastic wave map from the real line into the two-dimensional sphere, perturbed by multiplicative Gaussian noise of co-normal type. We prove that under this rescaling, the solutions converge to those of the deterministic heat flow for harmonic maps, revealing a transition from stochastic hyperbolic to deterministic parabolic dynamics. We further analyze the fluctuations around this limit, proving a weak central limit theorem and identifying the limiting process as the solution to a linear stochastic partial differential equation. The study combines tools from geometric analysis, stochastic calculus, and functional analysis, offering insights into the interplay between geometry, noise, and scaling in nonlinear stochastic systems.

math.PR↗
arXiv2025

Stochastic wave equations with constraints: well-posedness and Smoluchowski-Kramers diffusion approximation

We investigate the well-posedness of a class of stochastic second-order in time damped evolution equations in Hilbert spaces, subject to the constraint that the solution lie within the unitary sphere. Then, we focus on a specific example, the stochastic damped wave equation in a bounded domain of a $d$-dimensional Euclidean space, endowed with the Dirichlet boundary condition, with the added constraint that the $L^2$-norm of the solution is equal to one. We introduce a small mass $μ>0$ in front of the second-order derivative in time and examine the validity of a Smoluchowski-Kramers diffusion approximation. We demonstrate that, in the small mass limit, the solution converges to the solution of a stochastic parabolic equation subject to the same constraint. We further show that an extra noise-induced drift emerges, which in fact does not account for the Stratonovich-to-Itô correction term.

math.PR↗
arXiv2025

Nonlinear random perturbations of Reaction-Diffusion Equations

This paper investigates the well-posedness and small-noise asymptotics of a class of stochastic partial differential equations defined on a bounded domain of $\mathbb{R}^d$, where the diffusion coefficient depends nonlinearly and non-locally on the solution through a conditional expectation. The reaction term is assumed to be merely continuous and to satisfy a quasi-dissipativity condition, without requiring any growth bounds or local Lipschitz continuity. This setting introduces significant analytical challenges due to the temporal non-locality and the lack of regularity assumptions. Our results represent a substantial advance in the study of nonlinear stochastic perturbations of SPDEs, extending the framework developed in a previous paper.

math.PR↗
arXiv2024

The small-mass limit for some constrained wave equations with nonlinear conservative noise

We study the small-mass limit, also known as the Smoluchowski-Kramers diffusion approximation (see \cite{kra} and \cite{smolu}), for a system of stochastic damped wave equations, whose solution is constrained to live in the unitary sphere of the space of square-integrable functions on the interval $(0,L)$. The stochastic perturbation is given by a nonlinear multiplicative Gaussian noise, where the stochastic differential is understood in Stratonovich sense. Due to its particular structure, such noise not only conserves $\mathbb{P}$-a.s. the constraint, but also preserves a suitable energy functional. In the limit, we derive a deterministic system, that remains confined to the unit sphere of $L^2$, but includes additional terms. These terms depend on the reproducing kernel of the noise and account for the interaction between the constraint and the particular conservative noise we choose.

math.PR↗
arXiv2024

SPDEs on narrow channels and graphs: convergence and large deviations in case of non smooth noise

We investigate a class of stochastic partial differential equations of reaction-diffusion type defined on graphs, which can be derived as the limit of SPDEs on narrow planar channels. In the first part, we demonstrate that this limit can be achieved under less restrictive assumptions on the regularity of the noise, compared to [4]. In the second part, we establish the validity of a large deviation principle for the SPDEs on the narrow channels and on the graphs, as the width of the narrow channels and the intensity of the noise are jointly vanishing.

math.PR↗
arXiv2023

Smoluchowski-Kramers diffusion approximation for systems of stochastic damped wave equations with non-constant friction

We consider systems of damped wave equations with a state-dependent damping coefficient and perturbed by a Gaussian multiplicative noise. Initially, we investigate their well-posedness, under quite general conditions on the friction. Subsequently, we study the validity of the so-called Smoluchowski-Kramers diffusion approximation. We show that, under more stringent conditions on the friction, in the small-mass limit the solution of the system of stochastic damped wave equations converges to the solution of a system of stochastic quasi-linear parabolic equations. In this convergence, an additional drift emerges as a result of the interaction between the noise and the state-dependent friction. The identification of this limit is achieved by using a suitable generalization of the classical method of perturbed test functions, tailored to the current infinite dimensional setting.

math.PR↗
arXiv2023

On the small-mass limit for stationary solutions of stochastic wave equations with state dependent friction

We investigate the convergence, in the small mass limit, of the stationary solutions of a class of stochastic damped wave equations, where the friction coefficient depends on the state and the noisy perturbation if of multiplicative type. We show that the Smoluchowski-Kramers approximation that has been previously shown to be true in any fixed time interval, is still valid in the long time regime. Namely we prove that the first marginals of any sequence of stationary solutions for the damped wave equation converge to the unique invariant measure of the limiting stochastic quasilinear parabolic equation. The convergence is proved with respect to the Wasserstein distance associated with the $H^{-1}$ norm.

math.PR↗
arXiv2022

Averaging principle for slow-fast systems of stochastic PDEs with rough coefficients

In this paper, we consider a class of slow-fast systems of stochastic partial differential equations where the nonlinearity in the slow equation is not continuous and unbounded. We first provide conditions that ensure the existence of a martingale solution. Then we prove that the laws of the slow motions are tight, and any of their limiting points is a martingale solution for a suitable averaged equation. Our results apply to systems of stochastic reaction-diffusion equations where the reaction term in the slow equation is only continuous and has polynomial growth.

math.PR↗
arXiv2022

Nonlinear random perturbations of PDEs and quasi-linear equations in Hilbert spaces depending on a small parameter

We study a class of quasi-linear parabolic equations defined on a separable Hilbert space, depending on a small parameter in front of the second order term. Through the nonlinear semigroup associated with such equation, we introduce the corresponding SPDE and we study the asymptotic behavior of its solutions, depending on the small parameter. We show that a large deviations principle holds and we give an explicit description of the action functional.

math.PR↗
arXiv2020

Large deviations principle for the invariant measures of the 2D stochastic Navier-Stokes equations with vanishing noise correlation

We study the two-dimensional incompressible Navier-Stokes equation on the torus, driven by Gaussian noise that is white in time and colored in space. We consider the case where the magnitude of the random forcing $\sqrt{\e}$ and its correlation scale $δ(\e)$ are both small. We prove a large deviations principle for the solutions, as well as for the family of invariant measures, as $\e$ and $δ(\e)$ are simultaneously sent to $0$, under a suitable scaling.

math.PR↗
arXiv2020

Incompressible viscous fluids in $\mathbb{R}^2$ and SPDEs on graphs, in presence of fast advection and non smooth noise

The asymptotic behavior of a class of stochastic reaction-diffusion-advection equations in the plane is studied. We show that as the divergence-free advection term becomes larger and larger, the solutions of such equations converge to the solution of a suitable stochastic PDE defined on the graph associated with the Hamiltonian. Firstly, we deal with the case that the stochastic perturbation is given by a singular spatially homogeneous Wiener process taking values in the space of Schwartz distributions. As in previous works, we assume here that the derivative of the period of the motion on the level sets of the Hamiltonian does not vanish. Then, in the second part, without assuming this condition on the derivative of the period, we study a weaker type of convergence for the solutions of a suitable class of linear SPDEs.

math.PR↗