A large deviation principle for 2D stochastic Navier–Stokes equation

In this paper one specifies the ergodic behavior of the 2D-stochastic Navier–Stokes equation by giving a Large Deviation Principle for the occupation measure for large time. It describes the exact rate of exponential convergence. The considered random force is non-degenerate and compatible with the strong Feller property.     

Invariant measure for the stochastic Ginzburg Landau equation

The existence of martingale solutions and stationary solutions of stochastic Ginzburg-Landau equations under general hypothesizes on the dimension, the non linear term and the added noise is investigated. With a few more assumptions, it is established that the transition semi-group is well defined and that the stationary martingale solution yields the existence of an invariant measure. Moreover this invariant measure is shown to be unique.     

Computation of the invariant measure for a Lévy driven SDE: Rate of convergence

We study the rate of convergence of some recursive procedures based on some “exact” or “approximate” Euler schemes which converge to the invariant measure of an ergodic SDE driven by a Lévy process. The main interest of this work is to compare the rates induced by “exact” and “approximate” Euler schemes. In our main result, we show that replacing the small jumps by a Brownian component in the approximate case preserves the rate induced by the exact Euler scheme for a large class of Lévy processes.     

Ergodicity for a weakly damped stochastic non-linear Schrödinger equation

We study a damped stochastic non-linear Schr?dinger (NLS) equation driven by an additive noise. It is white in time and smooth in space. Using a coupling method, we establish convergence of the Markov transition semi-group toward a unique invariant probability measure. This kind of method was originally developed to prove exponential mixing for strongly dissipative equations such as the Navier-Stokes equations. We consider here a weakly dissipative equation, the damped nonlinear Schr?dinger equation in the one-dimensional cubic case. We prove that the mixing property holds and that the rate of convergence to equilibrium is at least polynomial of any power.     

A system of infinitely many mutually reffecting Brownian balls in ℝ d

Sumamry An infinite system of Skorohod type equations is studied. The unique solution of the system is obtained from a finite case by passing to the limit. It is a diffusion process describing a system of infinitely many Brownian hard balls and has a Gibbs state associated with the hard core pair potential as a reversible measure.On leave of, Department of Mathematics and Informatics, Faculty of Science Chiba University Chiba, 263 JapanSupported by Swiss National Foundation, contract Nr. 20-36305.92     

Large deviations for a reaction diffusion model

Summary We obtain large deviation estimates for the empirical measure of a class of interacting particle systems. These consist of a superposition of Glauber and Kawasaki dynamics and are described, in the hydrodynamic limit, by a reaction diffusion equation. We extend results of Kipnis, Olla and Varadhan for the symmetric exclusion process, and provide an approximation scheme for the rate functional. Some physical implications of our results are briefly indicated.     

Large deviations for Langevin spin glass dynamics

Summary We study the asymptotic behaviour of asymmetrical spin glass dynamics in a Sherrington-Kirkpatrick model as proposed by Sompolinsky-Zippelius. We prove that the annealed law of the empirical measure on path space of these dynamics satisfy a large deviation principle in the high temperature regime. We study the rate function of this large deviation principle and prove that it achieves its minimum value at a unique probability measureQ which is not markovian. We deduce that the quenched law of the empirical measure converges to _Q . Extending then the preceeding results to replicated dynamics, we investigate the quenched behavior of a single spin. We get quenched convergence toQ in the case of a symmetric initial law and even potential for the free spin.     

Homogenization and propagation of chaos to a nonlinear diffusion with sticky reflection

Summary We study a process reflecting in a domain. The process follows Wentzell non-sticky boundary conditions while being adsorbed at the boundary at a certain rate with respect to local time and desorbed at a rate with respect to natural time. We show that when the rates go to infinity with a converging ratio, the process converges to a process with sticky reflection having the limit ratio as the sojourn coefficient. We then study a mean-field interacting system of such particles. We show propagation of chaos to a nonlinear diffusion with sticky reflection when we perform this homogenization simultaneously as the number of particles goes to infinity.     

On the stochastic Benjamin-Ono equation

We discuss the Cauchy problem for the stochastic Benjamin-Ono equation in the function class Hs(R), s>3/2. When there is a zero-order dissipation, we also establish the existence of an invariant measure with support in H²(R). Many authors have discussed the Cauchy problem for the deterministic Benjamin-Ono equation. But our results are new for the stochastic Benjamin-Ono equation. Our goal is to extend known results for the deterministic equation to the stochastic equation.     

Boundary Harnack Principle for fractional powers of Laplacian on the Sierpiński carpet

We prove the Boundary Harnack Principle related to fractional powers of Laplacian for some natural regions in the two-dimensional Sierpiński carpet. This is a natural application of some more general approach based on the Ikeda-Watanabe formula, that expresses the harmonic measure in terms the Green function of a given region and the Lévy measure of the semigroup.     

Rates for the probability of large cubes being non-internally spanned in modified bootstrap percolation

Summary We find the exact rate of decay for the probability that a large cube is not internally spanned for the modified bootstrap percolation. It is proven that for cubes of large side the event that the cube is not internally spanned is essentially the same as the event that the cube possesses a completely vacant line.Research partially supported by NSF DMS 9157461 and a grant from the Sloan Foundation     

An Approximation for the Zakai Equation

In this article we consider a polygonal approximation to the unnormalized conditional measure of a filtering problem, which is the solution of the Zakai stochastic differential equation on measure space. An estimate of the convergence rate based on a distance which is equivalent to the weak convergence topology is derived. We also study the density of the unnormalized conditional measure, which is the solution of the Zakai stochastic partial differential equation. An estimate of the convergence rate is also given in this case. 60F25, 60H10.} Accepted 23 April 2001. Online publication 14 August 2001.     

Sur la mesure de sortie du super mouvement brownien

Summary We study some properties of the exit measure of super Brownian motion from a smooth domainD inR ^d . In particular, we give precise estimates for the probability that the exit measure gives a positive mass to a small ball on the boundary. As an application, we compute the Hausdorff dimension of the support of the exit measure. In dimension 2, we prove that the exit measure is absolutely continuous with respect to the Lebesgue measure on the boundary. In connection with Dynkin's work, our results give some information on the behavior of solutions of u=u ² inD, and are related to the characterization of removable singularities at the boundary. As a consequence of our estimates, we give a sufficient condition for the uniqueness of the positive solution of u=u ² inD that tends to on an open subsetO of D and to 0 on the complement in D of the closure ofO. Our proofs use the path-valued process studied in [L2, L3].

     

Nonlinear elliptic equations with measure data

In this paper we prove the existence of solutions of nonlinear equations of the type-div(a(x, u, Du)+H(x, u, Du)=f, wherea andH are Caratheodory functions andf is a bounded Radon measure. We remark that the operator can be not coercive. We give also some regularity results.     

Shock fluctuations in asymmetric simple exclusion

Summary The one dimensional nearest neighbors asymmetric simple exclusion process in used as a microscopic approximation to the Burgers equation. We study the process with rates of jumpsp>q to the right and left, respectively, and with initial product measure with densities < to the left and right of the origin, respectively (with shock initial conditions). We prove that a second class particle added to the system at the origin at time zero identifies microscopically the shock for all later times. If this particle is added at another site, then it describes the behavior of a characteristic of the Burgers equation. For vanishing left density (=0) we prove, in the scale t^1/2, that the position of the shock at timet depends only on the initial configuration in a region depending ont. The proofs are based on laws of large numbers for the second class particle.     

Asymptotic limit law for the close approach of two trajectories in expanding maps of the circle

Summary Given two pointsx, yS ¹ randomly chosen independently by a mixing absolutely continuous invariant measure of a piecewise expanding and smooth mapf of the circle, we consider for each >0 the point process obtained by recording the timesn>0 such that |f ⁿ (x)–f ⁿ (y)|. With the further assumption that the density of is bounded away from zero, we show that when tends to zero the above point process scaled by –1 converges in law to a marked Poisson point process with constant parameter measure. This parameter measure is given explicity by an average on the rate of expansion off.Partially supported by FAPESP grant number 90/3918-5     

Estimates for solutions of quasilinear problems with dead cores

A Rayleigh-Faber-Krahn type inequality is used to derive bounds for boundary value problems appearing in reaction-diffusion problems where the reactant is consumed. Interesting quantities are the minimum of the solution and the measure of the set where it vanishes. The proofs are rather elementary and apply to problems possessing solutions in a weak sense.     

Invariant measures for stochastic evolution equations of pure jump type

In this paper, we obtain a characterization of invariant measures of stochastic evolution equations and stochastic partial differential equations of pure jump type. As an application, it is shown that the equation has a unique invariant probability measure under some reasonable conditions.     

Removability of singularities in potential theory

IfQ is a compact metric space, a system of its closed subsets andg: R a prescribed nonnegative function, the conditions ong, and a closedF Q are specified guaranteeing the existence of a nontrivial Borel measure with support inF such that (L)g(L), L.For some kernels in potential theory these conditions permit to characterize geometrically those sets which contain support of a nontrivial measure whose potential belongs to a given class of functions. Several applications concerning removability of singularities of partial differential equations are presented.