Search results for " Probability"
showing 10 items of 2176 documents
A proof of Carleson's $\varepsilon^2$-conjecture
2019
In this paper we provide a proof of the Carleson $\varepsilon^2$-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated Carleson $\varepsilon^2$-square function.
Dimension of self-affine sets for fixed translation vectors
2018
An affine iterated function system is a finite collection of affine invertible contractions and the invariant set associated to the mappings is called self-affine. In 1988, Falconer proved that, for given matrices, the Hausdorff dimension of the self-affine set is the affinity dimension for Lebesgue almost every translation vectors. Similar statement was proven by Jordan, Pollicott, and Simon in 2007 for the dimension of self-affine measures. In this article, we have an orthogonal approach. We introduce a class of self-affine systems in which, given translation vectors, we get the same results for Lebesgue almost all matrices. The proofs rely on Ledrappier-Young theory that was recently ver…
Robustness of the Gaussian concentration inequality and the Brunn–Minkowski inequality
2016
We provide a sharp quantitative version of the Gaussian concentration inequality: for every $r>0$, the difference between the measure of the $r$-enlargement of a given set and the $r$-enlargement of a half-space controls the square of the measure of the symmetric difference between the set and a suitable half-space. We also prove a similar estimate in the Euclidean setting for the enlargement with a general convex set. This is equivalent to the stability of the Brunn-Minkowski inequality for the Minkowski sum between a convex set and a generic one.
Set-valued Brownian motion
2015
Brownian motions, martingales, and Wiener processes are introduced and studied for set valued functions taking values in the subfamily of compact convex subsets of arbitrary Banach space $X$. The present paper is an application of one the paper of the second author in which an embedding result is obtained which considers also the ordered structure of $ck(X)$ and f-algebras.
Contextuality in canonical systems of random variables
2017
Random variables representing measurements, broadly understood to include any responses to any inputs, form a system in which each of them is uniquely identified by its content (that which it measures) and its context (the conditions under which it is recorded). Two random variables are jointly distributed if and only if they share a context. In a canonical representation of a system, all random variables are binary, and every content-sharing pair of random variables has a unique maximal coupling (the joint distribution imposed on them so that they coincide with maximal possible probability). The system is contextual if these maximal couplings are incompatible with the joint distributions o…
Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs
2021
We introduce a decoupling method on the Wiener space to define a wide class of anisotropic Besov spaces. The decoupling method is based on a general distributional approach and not restricted to the Wiener space. The class of Besov spaces we introduce contains the traditional isotropic Besov spaces obtained by the real interpolation method, but also new spaces that are designed to investigate backwards stochastic differential equations (BSDEs). As examples we discuss the Besov regularity (in the sense of our spaces) of forward diffusions and local times. It is shown that among our newly introduced Besov spaces there are spaces that characterize quantitative properties of directional derivat…
A Laplace type problem for a lattice with cell composed by two trapezius and a triangle
2014
In the previous papers, [1], [2], [3], [4], [5], [6], [7], [8], [9] and [10] the authors study some Laplace problem for different lattices. In this paper we determine the probability that a random segment of constant length intersects a side of lattice with cell represented in fig.1
A Density Result for Homogeneous Sobolev Spaces on Planar Domains
2018
We show that in a bounded simply connected planar domain $\Omega$ the smooth Sobolev functions $W^{k,\infty}(\Omega)\cap C^\infty(\Omega)$ are dense in the homogeneous Sobolev spaces $L^{k,p}(\Omega)$.
Maximal regularity via reverse Hölder inequalities for elliptic systems of n-Laplace type involving measures
2008
In this note, we consider the regularity of solutions of the nonlinear elliptic systems of n-Laplacian type involving measures, and prove that the gradients of the solutions are in the weak Lebesgue space Ln,∞. We also obtain the a priori global and local estimates for the Ln,∞-norm of the gradients of the solutions without using BMO-estimates. The proofs are based on a new lemma on the higher integrability of functions.
On BLD-mappings with small distortion
2021
We show that every $$L$$ -BLD-mapping in a domain of $$\mathbb {R}^{n}$$ is a local homeomorphism if $$L < \sqrt{2}$$ or $$K_I(f) < 2$$ . These bounds are sharp as shown by a winding map.