Sufficient conditions for a real-valued Gaussian random field X = {X(t), t ∈ RN} with stationary increments to be strongly locally nondeterministic are proven. As applications, small ball probability estimates, Hausdorff measure of the sample paths, sharp Hölder conditions and tail probability estimates for the local times of Gaussian random fields are established. Running head: Strong Local No...

For a parabolic equation associated to a uniformly elliptic operator, we obtain a W 3,ε estimate, which provides a lower bound on the Lebesgue measure of the set on which a viscosity solution has a quadratic expansion. The argument combines parabolic W 2,ε estimates with a comparison principle argument. As an application, we show, assuming the operator is C1, that a viscosity solution is C2,α o...

1.1 Notations and Hypotheses In the sequel, Ω is a bounded domain in R (N ≥ 2), with a Lipschitz continuous boundary. n is the unit normal to ∂Ω outward to Ω. We denote by x · y the usual Euclidean product of two vectors (x, y) ∈ R × R ; the associated Euclidean norm is written |.|. The Lebesgue measure of a measurable subset E of R is denoted by |E|; σ is the Lebesgue measure on ∂Ω (i.e. the (...

In this paper, Ω is a bounded domain in R (N ≥ 2), with a Lipschitz continuous boundary. The unit normal to ∂Ω outward to Ω is denoted by n. We denote by x·y the usual Euclidean product of two vectors (x, y) ∈ R × R ; the associated Euclidean norm is written |.|. The Lebesgue measure of a measurable subset E in R is denoted by |E|; σ is the Lebesgue measure on ∂Ω (i.e. the (N−1)-dimensional Hau...

Many real phenomena may be modelled as random closed sets in Rd, of different Hausdorff dimensions. The authors have recently revisited the concept of mean geometric densities of random closed sets Θn with Hausdorff dimension n ≤ d with respect to the standard Lebesgue measure on Rd, in terms of expected values of a suitable class of linear functionals (Delta functions à la Dirac). In many real...

Following a recent paper [10] we show that the finiteness of square function associated with the Riesz transforms with respect to Hausdorff measure H implies that s is integer.

The �-dimensional Lebesgue space is the measurable space (E���(E�))— where E = [0 � 1) or E = R—endowed with the Lebesgue measure, and the “calculus of functions” on Lebesgue space is just “real and harmonic analysis.” The �-dimensional Gauss space is the same measure space (R���(R�)) as in the previous paragraph, but now we endow that space with the Gauss measure P� in place of the Lebesgue me...

This thesis combines computability theory and various notions of fractal dimension, mainly Hausdorff dimension. An algorithmic approach to Hausdorff measures makes it possible to define the Hausdorff dimension of individual points instead of sets in a metric space. This idea was first realized by Lutz (2000b). Working in the Cantor space 2ω of all infinite binary sequences, we study the theory ...

We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We announce the following results and explain some key ideas that go into their proofs. The thickness tend...