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Sandra Cerrai

Publications and source records attributed to Sandra Cerrai.

At least 37 records · Page 2Linked to original sources
arXiv2020

An averaging approach to the Smoluchowski-Kramers approximation in the presence of a varying magnetic field

We study the small mass limit of the equation describing planar motion of a charged particle of a small mass $μ$ in a force field, containing a magnetic component, perturbed by a stochastic term. We regularize the problem by adding a small friction of intensity $\e>0$. We show that for all small but fixed frictions the small mass limit of $q_{μ, \e}$ gives the solution $q_\e$ to a stochastic first order equation, containing a noise-induced drift term. Then, by using a generalization of the classical averaging theorem for Hamiltonian systems by Freidlin and Wentzell, we take the limit of the slow component of the motion $q_\e$ and we prove that it converges weakly to a Markov process on the graph obtained by identifying all points in the same connected components of the level sets of the magnetic field intensity function.

math.PR↗
arXiv2018

On the convergence of stationary solutions in the Smoluchowski-Kramers approximation of infinite dimensional systems

We prove the convergence, in the small mass limit, of statistically invariant states for a class of semi-linear damped wave equations, perturbed by an additive Gaussian noise, both with Lipschitz-continuous and with polynomial non-linearities. In particular, we prove that the first marginals of any sequence of invariant measures for the stochastic wave equation converge in a suitable Wasserstein metric to the unique invariant measure of the limiting stochastic semi-linear parabolic equation obtained in the Smoluchowski-Kramers approximation. The Wasserstein metric is associated to a suitable distance on the space of square integrable functions, that is chosen in such a way that the dynamics of the limiting stochastic parabolic equation is contractive with respect to such a Wasserstein metric. This implies that the limiting result is a consequence of the validity of a generalized Smoluchowski-Kramers limit at fixed times. The proof of such a generalized limit requires new delicate bounds for the solutions of the stochastic wave equation, that must be uniform with respect to the size of the mass.

math.PR↗
arXiv2018

Large deviations for fast transport stochastic RDEs with applications to the exit problem

We study reaction diffusion equations with a deterministic reaction term as well as two random reaction terms, one that acts on the interior of the domain, and another that acts only on the boundary of the domain. We are interested in the regime where the relative sizes of the diffusion and reaction terms are different. Specifically, we consider the case where the diffusion rate is much larger than the rate of reaction, and the deterministic rate of reaction is much larger than either of the random rate of reactions.

math.PR↗
arXiv2017

Large deviations for the dynamic $Φ^{2n}_d$ model

We are dealing with the validity of a large deviation principle for a class of reaction-diffusion equations with polynomial nonlinearity, perturbed by a Gaussian random forcing. We are here interested in the regime where both the strength of the noise and its correlation are vanishing, on a length scale $ε$ and $δ(ε)$, respectively, with $0<ε,δ(ε)<<1$. We prove that, under the assumption that $ε$ and $δ(ε)$ satisfy a suitable scaling limit, a large deviation principle holds in the space of continuous trajectories with values both in the space of square-integrable functions and in Sobolev spaces of negative exponent. Our result is valid, without any restriction on the degree of the polynomial nor on the space dimension.

math.PR↗
arXiv2016

Large deviations for the two-dimensional stochastic Navier-Stokes equation with vanishing noise correlation

We are dealing with the validity of a large deviation principle for the two-dimensional Navier-Stokes equation, with periodic boundary conditions, perturbed by a Gaussian random forcing. We are here interested in the regime where both the strength of the noise and its correlation are vanishing, on a length scale $\e$ and $\d(\e)$, respectively, with $0<\e,\ \d(\e)<<1$. Depending on the relationship between $\e$ and $\d(\e)$ we will prove the validity of the large deviation principle in different functional spaces.

math.PR↗
arXiv2016

Averaging principle for non autonomous slow-fast systems of stochastic RDEs: the almost periodic case

We study the validity of an averaging principle for a slow-fast system of stochastic reaction diffusion equations. We assume here that the coefficients of the fast equation depend on time, so that the classical formulation of the averaging principle in terms of the invariant measure of the fast equation is not anymore available. As an alternative, we introduce the time depending evolution family of measures associated with the fast equation. Under the assumption that the coefficients in the fast equation are almost periodic, the evolution family of measures is almost periodic. This allows to identify the appropriate averaged equation and prove the validity of the averaging limit.

math.PR↗
arXiv2016

On the Smoluchowski-Kramers approximation for SPDEs and its interplay with large deviations and long time behavior

We discuss here the validity of the small mass limit (the so-called Smoluchowski-Kramers approximation) on a fixed time interval for a class of semi-linear stochastic wave equations, both in the case of the presence of a constant friction term and in the case of the presence of a constant magnetic field. We also consider the small mass limit in an infinite time interval and we see how the approximation is stable in terms of the invariant measure and of the large deviation estimates and the exit problem from a bounded domain of the space of square integrable functions.

math.PR↗
arXiv2015

Large Deviations for the Langevin equation with strong damping

We study large deviations in the Langevin dynamics, with damping of order $\e^{-1}$ and noise of order $1$, as $\e\downarrow 0$. The damping coefficient is assumed to be state dependent. We proceed first with a change of time and then, we use a weak convergence approach to large deviations and their equivalent formulation in terms of the Laplace principle, to determine the good action functional. Some applications of these results to the exit problem from a domain and to the wave front propagation for a suitable class of reaction diffusion equations are considered.

math.PR↗
arXiv2014

SPDEs on narrow domains and on graphs: an asymptotic approach

We introduce here a class of stochastic partial differential equations defined on a graph and we show how they are obtained as the limit of suitable stochastic partial equations defined in a narrow channel, as the width of the channel goes to zero. To our knowledge, this is the first time an SPDE on a graph is studied.

math.PR↗
arXiv2014

On the Smoluchowski-Kramers approximation for a system with infinite degrees of freedom exposed to a magnetic field

We study the validity of the so-called Smoluchowski-Kramers approximation for a two dimensional system of stochastic partial differential equations, subject to a constant magnetic field. As the small mass limit does not yield to the solution of the corresponding first order system, we regularize our problem by adding a small friction. We show that in this case the Smoluchowski-Kramers approximation holds. We also give a justification of the regularization, by showing that the regularized problems provide a good approximation to the original ones.

math.PR↗
arXiv2014

Quasipotential and exit time for 2D Stochastic Navier-Stokes equations driven by space time white noise

We are dealing with the Navier-Stokes equation in a bounded regular domain $D$ of $\mathbb{R}^2$, perturbed by an additive Gaussian noise $\partial w^{Q_δ}/\partial t$, which is white in time and colored in space. We assume that the correlation radius of the noise gets smaller and smaller as $δ\searrow 0$, so that the noise converges to the white noise in space and time. For every $δ>0$ we introduce the large deviation action functional $S^δ_{0,T}$ and the corresponding quasi-potential $U_δ$ and, by using arguments from relaxation and $Γ$-convergence we show that $U_δ$ converges to $U=U_0$, in spite of the fact that the Navier-Stokes equation has no meaning in the space of square integrable functions, when perturbed by space-time white noise. Moreover, in the case of periodic boundary conditions the limiting functional $U$ is explicitly computed. Finally, we apply these results to estimate of the asymptotics of the expected exit time of the solution of the stochastic Navier-Stokes equation from a basin of attraction of an asymptotically stable point for the unperturbed system.

math.PR↗
arXiv2014

Smoluchowski-Kramers approximation and large deviations for infinite dimensional gradient systems

In this paper, we explicitly calculate the quasi-potentials for the damped semilinear stochastic wave equation when the system is of gradient type. We show that in this case the infimum of the quasi-potential with respect to all possible velocities does not depend on the density of the mass and does coincide with the quasi-potential of the corresponding stochastic heat equation that one obtains from the zero mass limit. This shows in particular that the Smoluchowski-Kramers approximation can be used to approximate long time behavior in the zero noise limit, such as exit time and exit place from a basin of attraction.

math.PR↗
arXiv2014

Smoluchowski-Kramers approximation and large deviations for infinite dimensional non-gradient systems with applications to the exit problem

In this paper, we study the quasi-potential for a general class of damped semilinear stochastic wave equations. We show that, as the density of the mass converges to zero, the infimum of the quasi-potential with respect to all possible velocities converges to the quasi-potential of the corresponding stochastic heat equation, that one obtains from the zero mass limit. This shows in particular that the Smoluchowski-Kramers approximation is not only valid for small time, but, in the zero noise limit regime, can be used to approximate long-time behaviors such as exit time and exit place from a basin of attraction.

math.PR↗
arXiv2012

A basic identity for Kolmogorov operators in the space of continuous functions related to RDEs with multiplicative noise

We consider the Kolmogorov operator associated with a reaction-diffusion equation having polynomially growing reaction coefficient and perturbed by a noise of multiplicative type, in the Banach space $E$ of continuous functions. By analyzing the smoothing properties of the associated transition semigroup, we prove a modification of the classical identité du carré di champs that applies to the present non-Hilbertian setting. As an application of this identity, we construct the Sobolev space $W^{1,2}(E;μ)$, where $μ$ is an invariant measure for the system, and we prove the validity of the Poincaré inequality and of the spectral gap.

math.AP↗
arXiv2012

Pathwise uniqueness for stochastic reaction-diffusion equations in Banach spaces with an Hölder drift component

We prove pathwise uniqueness for an abstract stochastic reaction-diffusion equation in Banach spaces. The drift contains a bounded Hölder term; in spite of this, due to the space-time white noise it is possible to prove pathwise uniqueness. The proof is based on a detailed analysis of the associated Kolmogorov equation. The model includes examples not covered by the previous works based on Hilbert spaces or concrete SPDEs.

math.AP↗