David Dereudre

Bio: David Dereudre is an academic researcher from university of lille. The author has contributed to research in topics: Brownian motion & Boolean model. The author has an hindex of 11, co-authored 54 publications receiving 523 citations. Previous affiliations of David Dereudre include University of Valenciennes and Hainaut-Cambresis & Lille University of Science and Technology.

Introduction to the theory of Gibbs point processes

Abstract: The Gibbs point processes (GPP) constitute a large class of point processes with interaction between the points. The interaction can be attractive, repulsive, depending on geometrical features whereas the null interaction is associated with the so-called Poisson point process. In a first part of this mini-course, we present several aspects of finite volume GPP defined on a bounded window in \(\mathbb {R}^d\). In a second part, we introduce the more complicated formalism of infinite volume GPP defined on the full space \(\mathbb {R}^d\). Existence, uniqueness and non-uniqueness of GPP are non-trivial questions which we treat here with completely self-contained proofs. The DLR equations, the GNZ equations and the variational principle are presented as well. Finally we investigate the estimation of parameters. The main standard estimators (MLE, MPLE, Takacs-Fiksel and variational estimators) are presented and we prove their consistency. For sake of simplicity, during all the mini-course, we consider only the case of finite range interaction and the setting of marked points is not presented.

Existence of Gibbsian point processes with geometry-dependent interactions

Abstract: We establish the existence of stationary Gibbsian point processes for interactions that act on hyperedges between the points. For example, such interactions can depend on Delaunay edges or triangles, cliques of Voronoi cells or clusters of k-nearest neighbors. The classical case of pair interactions is also included. The basic tools are an entropy bound and stationarity.

Existence of Gibbsian point processes with geometry-dependent interactions

Abstract: We establish the existence of stationary Gibbsian point processes for interactions that act on hyperedges between the points. For example, such interactions can depend on Delaunay edges or triangles, cliques of Voronoi cells or clusters of $k$-nearest neighbors. The classical case of pair interactions is also included. The basic tools are an entropy bound and stationarity.

Propagation of Gibbsianness for Infinite-dimensional Gradient Brownian Diffusions

Abstract: We study the (strong-) Gibbsian character on \(\mathbb{R}^{\mathbb{Z}^d}\) of the law at time t of an infinite- imensional gradient Brownian diffusion, when the initial distribution is Gibbsian

The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains

Abstract: We prove the existence of infinite-volume quermass-interaction processes in a general setting of nonlocally stable interaction and nonbounded convex grains No condition on the parameters of the linear combination of the Minkowski functionals is assumed The only condition is that the square of the random radius of the grain admits exponential moments for all orders Our methods are based on entropy and large deviation tools

An Introduction To The Theory Of Point Processes

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