Mesure extérieure

Résumé

La notion de mesure extérieure (ou mesure extérieure au sens de Carathéodory) est un concept, dû au mathématicien Constantin Carathéodory, qui généralise dans un cadre axiomatique une construction utilisée par Henri Lebesgue pour définir la mesure de Lebesgue des parties Lebesgue-mesurables de la droite réelle. Définition Soit X un ensemble. Une mesure extérieure sur X est une fonction définie sur l'ensemble de toutes les parties de X : ::\varphi: {\mathcal P}(X) \rightarrow [0, +\infty] qui vérifie les trois conditions suivantes : *L'ensemble vide est de mesure extérieure nulle : :: \varphi(\varnothing) = 0

Monotonie : Pour toutes parties A et B de X, :: A\subset B\quad\Rightarrow\quad\varphi(A) \leq \varphi(B).
Sous-additivité dénombrable : Pour toute suite (A_j){j\geq1} de parties de X, :: \varphi\left(\bigcup{j

Source officielle

https://fr.wikipedia.org/wiki/Mesure_ext%C3%A9rieure

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Let X = {X(t); t ∈ RN} be a (N,d) fractional Brownian motion in Rd of index H ∈ (0,1). We study the local time of X for all temporal dimensions N and spatial dimensions d for which local time exist. We obtain two main results : R1. If we denote by Lx(I) the local time of X at x on I ⊂ RN, then there exists a positive finite constant c such that mφ(X-1(0) ∩ [0,1]N) = c L0([0,1]N), where φ(r) = rN-dH (log log 1/r)dH/N and mφ(E) is the Hausdorff φ-measure of E. This solves the problem of the relationship between the local time and the exact Hausdorff measure of zero set for X. R2. We refine results of Xiao (1997) for the local times of (N,d) fractional Brownian motion. We prove the law of iterated logarithm and global Hölder condition for the local time for our process. These results establish interesting properties which were only partially proved in the literature. This literature began with the work of Taylor and Wendel (1966) and Perkins (1981) for the first result; Kesten (1965) and Perkins (1981) for the second on the Brownian motion. It continued with several works of which that of Xiao (1997) on the locally nondeterministic processes with stationnary increments including the (N,d) fractional Brownian motion. An intermediate result is found to solve the case N > 1. We generalize the result of Kasahara et al. (1999) on the tail probability of local time.

EPFL2008

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Improved stability of optimal traffic paths

Maria Colombo

Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree, and the nervous or cardiovascular systems. Given a flow (traffic path) that transports a given measure mu(-) onto a target measure mu(+), along a 1-dimensional network, the transportation cost per unit length is supposed in these models to be proportional to a concave power alpha is an element of (0, 1) of the intensity of the flow. In this paper we address an open problem in the book Optimal transportation networks by Bernot, Caselles and Morel and we improve the stability for optimal traffic paths in the Euclidean space R-d, with respect to variations of the given measures (mu(-), mu(+)), which was known up to now only for alpha > 1 - 1/d. We prove it for exponents alpha > 1 - 1/d-1 [in particular, for every alpha is an element of (0, 1) when d = 2], for a fairly large class of measures mu(+) and mu(-).

2018

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