Soft and Hard Wall in a Stochastic Reaction Diffusion Equation

Bertini, Lorenzo; Brassesco, Stella; Buttà, Paolo

doi:10.1007/s00205-008-0154-0

Cited by 19 publications

(29 citation statements)

References 16 publications

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“…Rigorous results in the spirit of Theorem 2.3 that have been an important guide toward our result and proof can be found in the works about long time fluctuations of phase boundaries treated for Cahn-Allen SPDEs with bistable symmetric potential in the small noise limit [8,7,3,17] and in the zero temperature limit of an interacting Brownian model [18]. In these cases the dynamics takes place close to a hyperbolic invariant manifold, which is not a limit cycle: it is a manifold of invariant solutions which is either R or an interval with boundaries ( [1,4] are also in the same class of problems), and the limit dynamics are diffusions.…”

Section: )mentioning

confidence: 99%

Small noise and long time phase diffusion in stochastic limit cycle oscillators

Giacomin

Poquet

Shapira

2018

Journal of Differential Equations

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We study the effect of additive Brownian noise on an ODE system that has a stable hyperbolic limit cycle, for initial data that are attracted to the limit cycle. The analysis is performed in the limit of small noise -that is, we modulate the noise by a factor ε ց 0 -and on a long time horizon. We prove explicit estimates on the proximity of the noisy trajectory and the limit cycle up to times exp cε −2 , c > 0, and we show both that on the time scale ε −2 the dephasing (i.e., the difference between noiseless and noisy system measured in a natural coordinate system that involves a phase) is close to a Brownian motion with constant drift, and that on longer time scales the dephasing dynamics is dominated, to leading order, by the drift. The natural choice of coordinates, that reduces the dynamics in a neighborhood of the cycle to a rotation, plays a central role and makes the connection with the applied science literature in which noisy limit cycle dynamics are often reduced to a diffusion model for the phase of the limit cycle.2010 Mathematics Subject Classification: 60H10, 34F05, 60F17, 82C31, 92B25

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Section: )mentioning

confidence: 99%

Small noise and long time phase diffusion in stochastic limit cycle oscillators

Giacomin

Poquet

Shapira

2018

Journal of Differential Equations

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“…It is shownthat the location of the phase boundary performs a Brownian motion. These results have been improved in a number of ways, notably to include small asymmetries that result in a drift for the arising diffusion process [7] and to deal with macroscopically finite volumes [4] (which introduce a repulsive effect approaching the boundary). Also the case of stochastic phase field equations has been considered [5].…”

Section: 5mentioning

confidence: 99%

“…As a matter of fact, in spite of the fact that our work deals directly with an interacting system, and not with an SPDE model, our approach is closer to the one in the SPDE literature. However, as we have already pointed out, a non negligible point is that we are forced to perform an analysis in distribution spaces, in fact Sobolev spaces with negative exponent, in contrast to the approach in the space of continuous functions in [8,15,7,4,5]. We point out that approaches to dynamical mean field type systems via Hilbert spaces of distribution has been already taken up in [14] but in our case the specific use of weighted Sobolev spaces is not only a technical tool, but it is intimately related to the geometry of the contractive invariant manifold M .…”

Section: 5mentioning

confidence: 99%

Synchronization and random long time dynamics for mean-field plane rotators

Bertini

Giacomin

Poquet

2013

Probab. Theory Relat. Fields

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We consider the natural Langevin dynamics which is reversible with respect to the mean-field plane rotator (or classical spin XY) measure. It is well known that this model exhibits a phase transition at a critical value of the interaction strength parameter {Mathematical expression}, in the limit of the number {Mathematical expression} of rotators going to infinity. A Fokker-Planck PDE captures the evolution of the empirical measure of the system as {Mathematical expression}, at least for finite times and when the empirical measure of the system at time zero satisfies a law of large numbers. The phase transition is reflected in the fact that the PDE for {Mathematical expression} above the critical value has several stationary solutions, notably a stable manifold-in fact, a circle-of stationary solutions that are equivalent up to rotations. These stationary solutions are actually unimodal densities parametrized by the position of their maximum (the synchronization phase or center). We characterize the dynamics on times of order {Mathematical expression} and we show substantial deviations from the behavior of the solutions of the PDE. In fact, if the empirical measure at time zero converges as {Mathematical expression} to a probability measure (which is away from a thin set that we characterize) and if time is speeded up by {Mathematical expression}, the empirical measure reaches almost instantaneously a small neighborhood of the stable manifold, to which it then sticks and on which a non-trivial random dynamics takes place. In fact the synchronization center performs a Brownian motion with a diffusion coefficient that we compute. Our approach therefore provides, for one of the basic statistical mechanics systems with continuum symmetry, a detailed characterization of the macroscopic deviations from the large scale limit-or law of large numbers-due to finite size effects. But the interest for this model goes beyond statistical mechanics, since it plays a central role in a variety of scientific domains in which one aims at understanding synchronization phenomena. © 2013 Springer-Verlag Berlin Heidelberg

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“…It also gives an idea about the fluctuations. To make it precise, we recall the notion of center of a front, already considered in [11,10,8] to study the front fluctuations for the Allen-Cahn and phase field equations. Motivation and properties can be found in those articles and some of the references therein.…”

Section: Stability Of Mmentioning

confidence: 99%

Front fluctuations for the stochastic Cahn–Hilliard equation

Bertini¹,

Brassesco²,

Buttà³

2015

Braz. J. Probab. Stat.

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We consider the Cahn-Hilliard equation in one space dimension, perturbed by the derivative of a space and time white noise of intensity ε 1 2 , and we investigate the effect of the noise, as ε → 0, on the solutions when the initial condition is a front that separates the two stable phases. We prove that, given γ < 2 3 , with probability going to one as ε → 0, the solution remains close to a front for times of the order of ε −γ , and we study the fluctuations of the front in this time scaling. They are given by a one dimensional continuous process, self similar of order 1 4 and non Markovian, related to a fractional Brownian motion and for which a couple of representations are given.2010 Mathematics Subject Classification. Primary 60H15, 35R60; secondary 82C24.

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Soft and Hard Wall in a Stochastic Reaction Diffusion Equation

Cited by 19 publications

References 16 publications

Small noise and long time phase diffusion in stochastic limit cycle oscillators

Small noise and long time phase diffusion in stochastic limit cycle oscillators

Synchronization and random long time dynamics for mean-field plane rotators

Front fluctuations for the stochastic Cahn–Hilliard equation

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