Convergence to equilibrium for finite Markov processes, with application to the Random Energy Model

We estimate the distance in total variation between the law of a finite state Markov process at time t, starting from a given initial measure, and its unique invariant measure. We derive upper bounds for the time to reach the equilibrium. As an example of application we consider a special case of finite state Markov process in random environment: the Metropolis dynamics of the Random Energy Model. We also study the process of the environment as seen from the process

On ergodic two-armed bandits

A device has two arms with unknown deterministic payoffs and the aim is to asymptotically identify the best one without spending too much time on the other. The Narendra algorithm offers a stochastic procedure to this end. We show under weak ergodic assumptions on these deterministic payoffs that the procedure eventually chooses the best arm (i.e., with greatest Cesaro limit) with probability one for appropriate step sequences of the algorithm. In the case of i.i.d. payoffs, this implies a "quenched" version of the "annealed" result of Lamberton, Pag\`{e}s and Tarr\`{e}s [Ann. Appl. Probab. 14 (2004) 1424--1454] by the law of iterated logarithm, thus generalizing it. More precisely, if $(\eta_{\ell,i})_{i\in \mathbb {N}}\in\{0,1\}^{\mathbb {N}}$, $\ell\in\{A,B\}$, are the deterministic reward sequences we would get if we played at time $i$, we obtain infallibility with the same assumption on nonincreasing step sequences on the payoffs as in Lamberton, Pag\`{e}s and Tarr\`{e}s [Ann. Appl. Probab. 14 (2004) 1424--1454], replacing the i.i.d. assumption by the hypothesis that the empirical averages $\sum_{i=1}^n\eta_{A,i}/n$ and $\sum_{i=1}^n\eta_{B,i}/n$ converge, as $n$ tends to infinity, respectively, to $\theta_A$ and $\theta_B$, with rate at least $1/(\log n)^{1+\varepsilon}$, for some $\varepsilon >0$. We also show a fallibility result, that is, convergence with positive probability to the choice of the wrong arm, which implies the corresponding result of Lamberton, Pag\`{e}s and Tarr\`{e}s [Ann. Appl. Probab. 14 (2004) 1424--1454] in the i.i.d. case.Comment: Published in at http://dx.doi.org/10.1214/10-AAP751 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotic expansions of Laplace integrals for quantum state tomography

Bayesian estimation of a mixed quantum state can be approximated via maximum likelihood (MaxLike) estimation when the likelihood function is sharp around its maximum. Such approximations rely on asymptotic expansions of multi-dimensional Laplace integrals. When this maximum is on the boundary of the integration domain, as it is the case when the MaxLike quantum state is not full rank, such expansions are not standard. We provide here such expansions, even when this maximum does not belong to the smooth part of the boundary, as it is the case when the rank deficiency exceeds two. These expansions provide, aside the MaxLike estimate of the quantum state, confidence intervals for any observable. They confirm the formula proposed and used without precise mathematical justifications by the authors in an article recently published in Physical Review A.Comment: 17 pages, submitte

Boundary stabilization and control of wave equations by means of a general multiplier method

We describe a general multiplier method to obtain boundary stabilization of the wave equation by means of a (linear or quasi-linear) Neumann feedback. This also enables us to get Dirichlet boundary control of the wave equation. This method leads to new geometrical cases concerning the "active" part of the boundary where the feedback (or control) is applied. Due to mixed boundary conditions, the Neumann feedback case generate singularities. Under a simple geometrical condition concerning the orientation of the boundary, we obtain a stabilization result in linear or quasi-linear cases

Liapunov Multipliers and Decay of Correlations in Dynamical Systems

The essential decorrelation rate of a hyperbolic dynamical system is the decay rate of time-correlations one expects to see stably for typical observables once resonances are projected out. We define and illustrate these notions and study the conjecture that for observables in $C^1$, the essential decorrelation rate is never faster than what is dictated by the {\em smallest} unstable Liapunov multiplier

Scale Invariant Cosmology

An attempt is made here to extend to the microscopic domain the scale invariant character of gravitation - which amounts to consider expansion as applying to any physical scale. Surprisingly, this hypothesis does not prevent the redshift from being obtained. It leads to strong restrictions concerning the choice between the presently available cosmological models and to new considerations about the notion of time. Moreover, there is no horizon problem and resorting to inflation is not necessary.Comment: TeX, 20 page

Multi-layered Spiking Neural Network with Target Timestamp Threshold Adaptation and STDP

Spiking neural networks (SNNs) are good candidates to produce ultra-energy-efficient hardware. However, the performance of these models is currently behind traditional methods. Introducing multi-layered SNNs is a promising way to reduce this gap. We propose in this paper a new threshold adaptation system which uses a timestamp objective at which neurons should fire. We show that our method leads to state-of-the-art classification rates on the MNIST dataset (98.60%) and the Faces/Motorbikes dataset (99.46%) with an unsupervised SNN followed by a linear SVM. We also investigate the sparsity level of the network by testing different inhibition policies and STDP rules