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We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form $- ϵ Δ + \nabla F (\cdot) \nabla$ on $ℝ^{d}$ or subsets of $ℝ^{d}$ , where $F$ is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of $F$ can be related, up to multiplicative errors that tend to one as $ϵ ↓ 0$ , to the capacities of suitably constructed sets. We show that these capacities...

Metastability in reversible diffusion processes II: precise asymptotics for small eigenvalues

Anton Bovier, Véronique Gayrard, Markus Klein (2005)

Journal of the European Mathematical Society

We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering processes with generators of the form $- ϵ Δ + \nabla F (\cdot) \nabla$ on $ℝ^{d}$ or subsets of $ℝ^{d}$ , where $F$ is a smooth function with finitely many local minima. Here we consider only the generic situation where the depths of all local minima are different. We show that in general the exponentially small part of the spectrum...

Metastability of the three dimensional Ising model on a torus at very low temperatures.

Ben Arous, Gérard, Cerf, Raphaël (1996)

Electronic Journal of Probability [electronic only]

Metastable behaviour of small noise Lévy-Driven diffusions

Peter Imkeller, Ilya Pavlyukevich (2008)

ESAIM: Probability and Statistics

We consider a dynamical system in $ℝ$ driven by a vector field -U', where U is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a Lévy noise of small intensity and such that the heaviest tail of its Lévy measure is regularly varying. We show that the perturbed dynamical system exhibits metastable behaviour i.e. on a proper time scale it reminds of a Markov jump process taking values in the local minima of the potential U. Due to the heavy-tail nature...

Method of Ritz for random eigenvalue problems

Martin Hála (1994)

Kybernetika

Méthodes de noyaux et de symboles en calcul stochastique non commutatif

Dermoune, A. (1992)

Portugaliae mathematica

Méthodes probabilistes pour la détermination de résolvantes sous-markoviennes

François Bronner (1975)

Annales de l'I.H.P. Probabilités et statistiques

Méthodes stochastiques pour la détermination de polynômes de Bernstein

Serge Dubuc (1975)

Annales de l'I.H.P. Probabilités et statistiques

Metric diophantine approximation and probability.

Hensley, Doug (1998)

The New York Journal of Mathematics [electronic only]

Metric Diophantine approximation on the middle-third Cantor set

Yann Bugeaud, Arnaud Durand (2016)

Journal of the European Mathematical Society

Let $μ \geq 2$ be a real number and let $ℳ (μ)$ denote the set of real numbers approximable at order at least $μ$ by rational numbers. More than eighty years ago, Jarník and, independently, Besicovitch established that the Hausdorff dimension of $ℳ (μ)$ is equal to $2 / μ$ . We investigate the size of the intersection of $ℳ (μ)$ with Ahlfors regular compact subsets of the interval $[0, 1]$ . In particular, we propose a conjecture for the exact value of the dimension of $ℳ (μ)$ intersected with the middle-third Cantor set and give several results...

Metric entropy and the central limit theorem in $C (S)$

R. M. Dudley (1974)

Annales de l'institut Fourier

Central limit theorems with hypotheses in terms of $ϵ$ -entropy are proved first in $C (S)$ where $S$ is a compact metric space and then in an arbitrary separable Banach space.

Metric entropy of convex hulls in Hilbert spaces

Wenbo Li, Werner Linde (2000)

Studia Mathematica

Let T be a precompact subset of a Hilbert space. We estimate the metric entropy of co(T), the convex hull of T, by quantities originating in the theory of majorizing measures. In a similar way, estimates of the Gelfand width are provided. As an application we get upper bounds for the entropy of co(T), $T = t_{1}, t_{2}, . . .$ , $| | t_{j} | | \leq a_{j}$ , by functions of the $a_{j}$ ’s only. This partially answers a question raised by K. Ball and A. Pajor (cf. [1]). Our estimates turn out to be optimal in the case of slowly decreasing sequences ${(a_{j})}_{j = 1}^{\infty}$ .

Metric projections and best approximants in Bochner-Orlicz spaces.

Ryszard Pluciennik, Yuwen Wang (1994)

Revista Matemática de la Universidad Complutense de Madrid

In the first section of this paper there are given criteria for strict convexity and smoothness of the Bochner-Orlicz space with the Orlicz norm as well as the Luxemburg norm. In the second one that geometrical properties are applied to the characterization of metric projections and zero mean valued best approximants to Bochner-Orlicz spaces.

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Mesures invariantes sur des ensembles autosimilaires

Mesures singulières associées à des découpages aléatoires d'un hypercube

Mesures vectorielles dans les espaces réticulés

Metastability and the Ising model.

Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times