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In this contribution, we study the notion of affine invariance (specifically, invariance to the shifting, scaling, and rotation of the coordinate system) as a starting point for the development of mathematical tools and approaches useful in the characteriz ...
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) th ...
We study the simple random walk on the n-dimensional hypercube, in particular its hitting times of large (possibly random) sets. We give simple conditions on these sets ensuring that the properly rescaled hitting time is asymptotically exponentially distri ...
We study the local times of fractional Brownian motions for all temporal dimensions, N, spatial dimensions d and Hurst parameters H for which local times exist. We establish a Holder continuity result that is a refinement of Xiao (Probab Th Rel Fields 109: ...
We analyze the distribution of the distances between tributaries of a given size (or of sizes larger than a given area) draining along either an open boundary or the mainstream of a river network. By proposing a description of the distance separating presc ...
The subject of the present thesis is an optimal prediction problem concerning the ultimate maximum of a stable Lévy process over a finite interval of time. Such "optimal prediction" problems are of both theoretical and practical interest, in particular the ...
This thesis describes measurements of Brownian motion of a colloidal particle using optical trapping. Two aspects are investigated: (i) influence of inertial effects on Brownian motion, and (ii) effect of the optical trap on Brownian motion. The first part ...
Thermal position fluctuations of a colloidal particle in an optical trap are measured with microsecond resolution using back-focal-plane interferometry. The mean-square displacement MSD(t) and power spectral density are in excellent agreement with the theo ...
We consider multiple description coding for the Gaussian source with K descriptions under the symmetric mean squared error distortion constraints. One of the main contributions is a novel lower bound on the sum rate, derived by generalizing Ozarow’s well-k ...
We show that for a voter model on {0,1}Z corresponding to a random walk with kernel p(·) and starting from unanimity to the right and opposing unanimity to the left, a tight interface between 0's and 1's exists if p(·) has second moments but does not if p( ...
