# Month: April 2011

La correction de copies est sans doute l’activité la plus désagréable du beau métier d’enseignant. Les barèmes alambiqués compliquent grandement la tâche, et alimentent le mythe de la juste note (quelque soit l’injuste sujet !). Pour fabriquer un sujet d’examen facile à corriger, il suffit de préparer vingt questions, chacune valant 1 point, quelqu’en soit la difficulté. Notons que  cette pondération non linéaire rabote les extrêmes, ce qui réduit les illusions des bons et le désespoir des mauvais.

Saviez-vous que le mot barème vient de François Barrême (Tarascon 1638 – Paris 1703), fameux arithméticien français du XVIIe siècle considéré comme l’un des fondateurs de la comptabilité ? Il fait donc partie du club des éponymes aux côtés de Eugène Poubelle, Étienne de Silhouette, Félix KirJohn Montagu 4e comte de Sandwich, John Loudon McAdam, et tant d’autres.

Le 14 décembre 2009, à l’occasion de la présentation des priorités financées par l’Emprunt national, le Président de la République Nicolas Sarkozy annonça le changement de nom du CEA : « Il s’agit bien ici de respecter l’engagement du Gouvernement d’une parité absolue des efforts de recherche entre le nucléaire et les énergies renouvelables. Je vous annonce donc que pour porter cette nouvelle ambition, le CEA deviendra le Commissariat à l’Énergie Atomique et aux Énergies Alternatives ». Le CEA est donc Fukushima-compatible depuis plus d’un an. Étonnant.

Coq à l’âne (sans mauvais jeu de mots) : Jean CÉA.

Few days ago Arnaud Guillin uploaded the joint paper arXiv:1104.2198 with Patrick Cattiaux entitled Central limit theorems for additive functionals of ergodic Markov diffusions processes. In this paper, we revisit functional central limit theorems for additive functionals of ergodic Markov diffusion processes. Translated in the language of partial differential equations of evolution, they appear as diffusion limits in the asymptotic analysis of Fokker-Planck type equations. We focus on the square integrable framework, and we provide tractable conditions on the infinitesimal generator, including degenerate or anomalously slow diffusions. We take advantage on recent developments in the study of the trend to the equilibrium of ergodic diffusions via functional inequalities and Lyapunov criteria. We discuss examples and formulate open problems.

Let ${{(X_t)}_{t\geq0}}$ be a continuous time strong Markov process with state space ${\mathbb{R}^d}$, non explosive, irreducible, positive recurrent, with unique invariant probability measure ${\mu}$. For every ${f\in\mathbb{L}^1(\mu)}$, if almost surely (a.s.) the function ${s\in\mathbb{R}_+\mapsto f(X_s)}$ is locally Lebesgue integrable, then

$\frac{S_t}{t} \underset{t\rightarrow\infty}{\overset{\text{a.s.}}{\longrightarrow}} \int\!f\,d\mu \quad\text{where}\quad S_t:=\int_0^t\!f(X_s)\,ds. \ \ \ \ \ (1)$

If ${X_0\sim\mu}$ then by the Fubini theorem (1) holds for all ${f\in\mathbb{L}^1(\mu)}$ and the convergence holds additionally in ${\mathbb{L}^1}$ thanks to the dominated convergence theorem. The statement (1) which relates an average in time with an average in space is an instance of the ergodic phenomenon. It can be seen as a strong law of large numbers for the additive functional ${{(S_t)}_{t\geq0}}$ of the Markov process ${{(X_t)}_{t\geq0}}$. The asymptotic fluctuations are described by a central limit theorem which is the subject of this work. Let us assume that ${X_0\sim\mu}$ and ${f\in\mathbb{L}^2(\mu)}$ with ${\int\!f\,d\mu=0}$ and ${f\neq0}$. Then for all ${t\geq0}$ we have ${S_t\in\mathbb{L}^2(\mu)\subset\mathbb{L}^1(\mu)}$ and ${\mathbb{E}(S_t)=0}$. We say that ${{(S_t)}_{t\geq0}}$ satisfies to a central limit theorem (CLT) when

$\frac{S_t}{s_t} \overset{\text{law}}{\underset{t\rightarrow\infty}{\longrightarrow}} \mathcal{N}(0,1) \ \ \ \ \ (2)$

for a deterministic positive function ${t\mapsto s_t}$ which may depend on ${f}$. Here ${\mathcal{N}(0,1)}$ stands for the standard Gaussian law on ${\mathbb{R}}$ with mean ${0}$ and variance ${1}$. By analogy with the CLT for i.i.d. sequences one may expect that ${s_t^2=\mathrm{Var}(S_t)}$ and that this variance is of order ${t}$ as ${t\rightarrow\infty}$. A standard strategy for proving (2) consists in representing ${{(S_t)}_{t\geq0}}$ as a sum of an ${\mathbb{L}^2}$-martingale plus a remainder term which vanishes in the limit, reducing the proof to a central limit theorem for martingales. This strategy is particularly simple under mild assumptions, see for instance the book of Jacod and Shiryaev (VII.3 page 486). Namely, if ${L}$ is the infinitesimal generator of ${{(X_t)}_{t\geq0}}$ with domain ${\mathbb{D}(L)\subset\mathbb{L}^2(\mu)}$ and if ${g\in\mathbb{D}(L)}$ then ${{(M_t)}_{t\geq0}}$ defined by

$M_t:=g(X_t)-g(X_0)-\int_0^t\!(Lg)(X_s)\,ds$

is a local ${\mathbb{L}^2}$ martingale. Now if ${g^2\in\mathbb{D}(L)}$ and

$\Gamma(g):=L(g^2)-2gLg\in\mathbb{L}^1(\mu)$

then

$\left<M\right>_t=\int_0^t\!\Gamma(g)(X_s)\,ds.$

The law of large numbers (1) yields

$\lim_{t\rightarrow\infty}\frac{1}{t}\left<M\right>_t=\int\!\Gamma(g)\,d\mu.$

As a consequence, for a prescribed ${f}$, if the Poisson equation

$Lg=f$

admits a mild enough solution ${g}$ then

$\frac{M_t}{s_t}=\frac{g(X_t)-g(X_0)}{s_t}-\frac{S_t}{s_t}.$

This suggests to deduce (2) from a CLT for martingales (survey paper by Ward). We begin by revisiting this strategy. Beyond (2), we say that ${{(S_t)}_{t\geq0}}$ satisfies to a Functional Central Limit Theorem (FCLT) or Invariance Principle when for every finite sequence ${0<t_1\leq\cdots\leq t_n<\infty}$,

$\left(\frac{S_{t_1/\varepsilon}}{s_{t_1/\varepsilon}}, \ldots,\frac{S_{t_n/\varepsilon}}{s_{t_n/\varepsilon}}\right) \overset{\text{law}}{\underset{\varepsilon\rightarrow0}{\longrightarrow}} \mathcal{L}\left(\left(B_{t_1},\ldots,B_{t_n}\right)\right) \ \ \ \ \ (3)$

where ${{(B_t)}_{t\geq0}}$ is a standard Brownian Motion on ${\mathbb{R}}$. Taking ${n=1}$ gives (2). To capture multitime correlations, one may upgrade the convergence in law in (3) to an ${\mathbb{L}^2}$ convergence. The statement (3) means that as ${\varepsilon\rightarrow0}$, the rescaled process ${{(S_{t/\varepsilon}/s_{t/\varepsilon})}_{t\geq0}}$ converges in law to a Brownian Motion, for the topology of finite dimensional marginal laws. At the level of Chapman-Kolmogorov-Fokker-Planck equations, (3) is a diffusion limit for a weak topology.

We focus on the case where ${{(X_t)}_{t\geq0}}$ is a Markov diffusion process on ${E=\mathbb{R}^d}$, and we seek for conditions on ${f}$ and on the infinitesimal generator in order to get (2) or even (3). We revisit the renowned result of Kipnis and Varadhan, and provide an alternative approach which is not based on the resolvent. Our results cover fully degenerate situations such as kinetic models. More generally, we believe that a whole category of diffusion limits which appear in the asymptotic analysis of evolution partial differential equations of Fokker-Planck type enters indeed the framework of the central limit theorems we shall discuss. We also explain how the behavior out of equilibrium (i.e. ${X_0\not\sim\mu}$) may be recovered from the behavior at equilibrium (i.e. ${X_0\sim\mu}$) by using propagation of chaos (decorrelation), for instance via Lyapunov criteria ensuring a quick convergence in law of ${X_t}$ to ${\mu}$ as ${t\rightarrow\infty}$. Note that since we focus on an ${\mathbb{L}^2}$ framework, the natural normalization is the square root of the variance and we can only expect Gaussian fluctuations. We believe however that stable limits that are not Gaussian, also known as “anomalous diffusion limits”, can be studied using similar tools, see for instance the forthcoming book Fluctuations in Markov Processes by Landim, Olla, and Komorowski.

Voici ce que m’inspire le prix de l’immobilier dans le grand Paris. Dans mon quartier, à Vincennes, nous en sommes déjà au billet vert… Autre idée amusante : le prix au kilo des pièces de monnaie.

RectoVersoValeurDimensions€/m²
5 €120 × 62672
10 €127 × 671175
20 €133 × 722089
50 €140 × 774638
100 €147 × 828296
200 €153 × 8215941
500 €160 × 8238110

Les dimensions et les images proviennent de Wikipédia, comme en témoignent les liens.

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