20th Meeting of AiOs in Stochastics

The twentieth meeting of the AiOs (PhD students) in Stochastics in the Netherlands will take place from 21 - 23  May 2012. This meeting is organized by the AiO Network Stochastics and is supported by The Dutch Research School in Mathematics WONDER and by the mathematics cluster NDNS+, and stochastics cluster STAR. The meetings continue the tradition of AiO meetings following the Bijeenkomst Stochastici in Lunteren, at a different location, a different time,  and in a different format.

The meeting consists of two short courses, one in probability and one in statistics, and lectures by the participants. The purpose is both to learn topics of current research interest and to become acquainted with the work of other aios in the Netherlands.
 

Programme
The short courses are given by prof. T. Gneiting (Heidelberg, Dld) and prof. F.M. Dekking (Delft). See below for titles and abstracts.
The meeting will start on Monday at 10.30 and will end on Wednesday after lunch. For a detailed programme see below.

Location
The meeting will be held in  building "Stalheim" (see picture)  of
 Conferentie Centrum De Hoorneboeg
 Hoorneboeg 5
 1213 RE Hilversum
  035 - 577 1231

Registration
Registration is possible electronically through the registration form.

Conference Fee
The total fee for the conference and two nights overnight stay in building Stalheim and meals from Monday lunch to Wednesday lunch is €235,50. See the registration form for specifics on payment.

Further Information
For further information contact the organizers Rob van den Berg and Geurt Jongbloed, or Maryke Titawano for practical matters.
 
 
 

Detailed Program
 

Abstracts and Titles

Tilmann Gneiting, University of Heidelberg
Spatial Statistics, Weather and Climate
Abstract: I will discuss the correlation theory of isotropic stochastic processes on Euclidean domains and spheres, which offers a wide range of challenging open problems. Applications and case studies in weather and climate research call for an increased involvement of probabilists and statisticians in the atmospheric sciences.

Michel Dekking, Delft
Random Fractals
Abstract: We first introduce the notion of Hausdorff dimension. Then we define random Cantor sets by means of randomly labeled trees. The canonical example is Mandelbrot percolation, also called fractal percolation. We discuss its properties, and show in particular that when we move the density parameter, then there is a non-trivial phase transition from almost sure total disconnectedness to very large connected components occurring with a positive probability. The last part of the course considers graphs of functions. We establish the connection between Hölder continuity and Hausdorff dimension, introduce the energy method, and end with computing the almost sure dimension of the graph of Brownian motion. Prerequisites: basic measure theory, as the construction of Borel measures and image measures, and e.g., the theorems of Dominated Convergence, Fubini, and the Borel-Cantelli Lemma.

Tim van de Brug, VUA
Fat fractal percolation and k-fractal percolation
Abstract: We consider two variations on the Mandelbrot fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided in N^d equal subcubes, k of which are retained while the others are discarded. The procedure is then iterated inside the retained cubes at all smaller scales. We show that the (properly rescaled) percolation critical value of this model converges to the critical value of ordinary site percolation on a particular d-dimensional lattice as N tends to infinity. This is analogous to the result of Falconer and Grimmett that the critical value for Mandelbrot fractal percolation converges to the critical value of site percolation on the same lattice. In the fat fractal percolation model, subcubes are retained with probability p_n at step n of the construction, where (p_n) is a non-decreasing sequence with \prod p_n > 0. The Lebesgue measure of the limit set is positive a.s. given non-extinction. We prove that either the set of connected components larger than one point has Lebesgue measure zero a.s. or its complement in the limit set has Lebesgue measure zero a.s. Joint work with Erik Broman, Federico Camia, Matthijs Joosten and Ronald Meester.

Dirk Erhard, Leiden
The parabolic Anderson model in a dynamic random environment
Abstract: The parabolic Anderson model in a dynamic random environment is a differential equation, which describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate $2d\kappa$, $\kappa > 0$, split into two and die at rates determined by the environment. We denote by u(x,t) the mean number of particles at site x at time t conditioned on the evolution of the environment. My main object of interest is the quenched Lyapunovexponent $\lambda(\kappa)= \lim_{t \to \infty} 1/t \log u(0,t)$. In this talk I will discuss some results concerning its qualitative behaviour as a function of $\kappa$.

Marcin Lis, VUA
Correlation functions in the 2D Ising model via signed loops and paths
Abstract: Using the combinatorial method for the 2D Ising model originating in the works of Sherman, Burgoyne and others we derive formulas for the correlation functions in terms of signed loops and paths. In the case of regular lattices we also identify the critical temperature for the phase transition in the long range behaviour of these functions. Joint work with Wouter Kager and Ronald Meester.

Kimberly McGarrity, Delft
Nonparametric Inference in a Stereological Model with Randomly Sized Oriented Cylinders
Abstract: We use oriented circular cylinders in an opaque medium to represent certain microstructural objects in steel. The opaque medium is sliced and on the cut plane parallel to the cylinder axes are the observable rectangular projections of the cylinders. There is a well established inverse relation between the dis- tribution of the observed 2D rectangle lengths and the distribution of the 3D cylinder radii that dates back to Wicksell (1925). Because the cylinders are oriented in our model, all of the height information for a given, cut cylinder is preserved. We propose a nonparametric estimation procedure to estimate the joint distribution of the 3D cylinder radii and heights from the observed 2D rectangle lengths and heights. Also, from the 2D observations, other interesting distributional properties of these cylinders are estimated, such as the covariance between the squared radii and heights, the distributions of the Aspect Ratio, Surface Area and Volume of the cylinders. Many of these quantities are directly linked to the mechanical properties of the material, and are, therefore, useful for industry. Finally, the mathematical model and estimation procedures are applied to two banded microstructures for which nearly 90 ?m of depth have been observed via serial sectioning.

Christos Pelekis, Delft
A problem on colored coin tosses
Abstract: Fix a positive integer d and color d fair coins with d colors under the restriction that no coin uses the same color on both of its sides. Toss the coins and let X_d be the number of different colors that we see after the toss. How does a coloring on the coins which maximizes the median of X_d  look like ? I will present some results on the problem and dicuss it's relation with a combinatorial question. This is joint work with Robbert Fokkink.

Renato Soares dos Santos, Leiden
Random walk in a supercritical contact process
Abstract: Random walks in dynamic random environments are models for diffusion in a disordered, evolving medium, which is usually assumed to be Markovian in time. While several results about the asymptotic behavior of the random walk are available when the medium has sufficiently strong and uniform mixing in space-time, much less is understood when mixing is non-uniform. We will discuss the case of the supercritical contact process, which is an example of this situation.