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.     

How regular can the boundary of a quadratic Siegel disk be?

In the family of quadratic polynomials with an irrationally indifferent fixed point, we show the existence of Siegel disks with a fine control on the degree of regularity of the linearizing map on their boundary. A general theorem is stated and proved. As a particular case, we show that in the quadratic family, there are Siegel disks whose boundaries are but not Jordan curves.

     

An adaptive enrichment algorithm for advection‐dominated problems

We are interested in developing a numerical framework well suited for advection–diffusion problems when the advection part is dominant. In that case, given Dirichlet type boundary condition, it is well known that a boundary layer develops. To resolve correctly this layer, standard methods consist in increasing the mesh resolution and possibly increasing the formal accuracy of the numerical method. In this paper, we follow another path: we do not seek to increase the formal accuracy of the scheme but, by a careful choice of finite element, to lower the mesh resolution in the layer. Indeed the finite element representation we choose is locally the sum of a standard one plus an enrichment. This paper proposes such a method and with several numerical examples, we show the potential of this approach. In particular, we show that the method is not very sensitive to the choice of the enrichment and develop an adaptive algorithm to automatically choose the enrichment functions.Copyright © 2012 John Wiley & Sons, Ltd.     

Simulated likelihood inference for stochastic volatility models using continuous particle filtering

Discrete-time stochastic volatility (SV) models have generated a considerable literature in financial econometrics. However, carrying out inference for these models is a difficult task and often relies on carefully customized Markov chain Monte Carlo techniques. Our contribution here is twofold. First, we propose a new SV model, namely SV–GARCH, which bridges the gap between SV and GARCH models: it has the attractive feature of inheriting unconditional properties similar to the standard GARCH model but being conditionally heavier tailed. Second, we propose a likelihood-based inference technique for a large class of SV models relying on the recently introduced continuous particle filter. The approach is robust and simple to implement. The technique is applied to daily returns data for S&P 500 and Dow Jones stock price indices for various spans.     

Quantitative Concentration Inequalities for Empirical Measures on Non-compact Spaces

We establish quantitative concentration estimates for the empirical measure of many independent variables, in transportation distances. As an application, we provide some error bounds for particle simulations in a model mean field problem. The tools include coupling arguments, as well as regularity and moment estimates for solutions of certain diffusive partial differential equations.     

<Emphasis Type="Italic">m</Emphasis>-Dissipativity of Kolmogorov Operators Corresponding to Burgers Equations with Space-time White Noise

In this article, we prove new a priori estimates on the solutions of the Burgers equation driven by a space-time white noise and on the associated invariant measure. We also prove smoothing properties for the transition semi-group. This is obtained thanks to the introduction of a modified Kolmogorov operator. These results are then used to prove that the Kolmogorov operator associated to the Burgers equation is m-dissipative. This implies several properties on the Kolmogorov equation.      

Monte-Carlo simulation of laser diode sub-Poissonian light generation

When laser diodes are driven by high-impedance electrical sources, the variance of the number of photo-detection events counted over large time durations is less than the average number of events (sub-Poissonian light). This paper presents a Monte-Carlo simulation that keeps track of each level occupancy (0 or 1) in the conduction and valence bands, and of the number of light quanta in the optical cavity. When there is good electron–lattice thermal contact, the electron and hole temperatures remain equal to that of the lattice. In that case, the elementary laser-diode noise theory results are accurately reproduced by the simulation. But when the thermal contact is poor (or, almost equivalently, at high power levels), new effects occur (spectral-hole burning, temperature fluctuations, statistical fluctuations of the optical gain) that are difficult to handle theoretically. Our numerical simulation shows that the frequency domain over which the photo-current spectral density is below the shot-noise level becomes narrower as the optical power increases.     

Partition of Waves in a System of Conservation Laws

The aim of this paper is to give a simple proof of the classical Liu estimate on the decay of positive waves in a solution of a n×n system of conservation laws. In the first part, we transcribe the wave partition technique introduced in Comm. Math. Phys. 57 (1977), 135–148 (by means of the Glimm scheme) to the case of approximate solutions constructed by the wave front tracking scheme. Then, we use a decoupling argument on the characteristic speeds to establish the desired estimate.