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La marche aléatoire (ou marche au hasard) est un objet fondamental de la théorie des probabilités. Un des problèmes les plus intéressants pour la marche aléatoire (ainsi que pour le mouvement brownien, son analogue dans un contexte continu) est de savoir comment elle recouvre des ensembles où se trouvent les points qui sont souvent (ou au contraire, rarement) visités, et combien il y a de tels points. Les travaux de Dembo, Peres, Rosen et Zeitouni permettent de résoudre plusieurs conjectures importantes...
Un théorème classique exprime qu’à partir d’un semi-groupe d’opérateurs sur l’espace des fonctions continues tendant vers 0 à l’infini, , , , continue, , on peut construire un processus markovien “standard”, à trajectoires réglées et continues à droite, quasi-continu à gauche ; l’espace des états est supposé localement compact à base dénombrable d’ouverts. Nous supposons ici que l’espace des états est seulement universellement mesurable dans un souslinien complètement régulier ; le processus...
Given two measure spaces equipped with liftings or densities (complete if liftings are considered) the existence of product liftings and densities with lifting invariant or density invariant sections is investigated. It is proved that if one of the marginal liftings is admissibly generated (a subclass of consistent liftings), then one can always find a product lifting which has the property that all sections determined by one of the marginal spaces are lifting invariant (Theorem 2.13). For a large...
In this article we formalize in Mizar [5] product pre-measure on product sets of measurable sets. Although there are some approaches to construct product measure [22], [6], [9], [21], [25], we start it from σ-measure because existence of σ-measure on any semialgebras has been proved in [15]. In this approach, we use some theorems for integrals.
We use the Calderón Maximal Function to prove the Kato-Ponce Product Rule Estimate and the Christ-Weinstein Chain Rule Estimate for the Hajłasz gradient on doubling measure metric spaces.
In the present article we provide an example of two closed non--lower porous sets such that the product is lower porous. On the other hand, we prove the following: Let and be topologically complete metric spaces, let be a non--lower porous Suslin set and let be a non--porous Suslin set. Then the product is non--lower porous. We also provide a brief summary of some basic properties of lower porosity, including a simple characterization of Suslin non--lower porous sets in topologically...
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