Quermass-interaction processes: Conditions for stability

We consider a class of random point and germ-grain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places a grain, a (possibly random) compact convex set. Let Ξ be the union of all grains. One can now construct new processes whose density is derived from an exponential of a linear combination of quermass functionals of Ξ. If only the area functional is used, then the area-interaction point process is recovered. New point processes arise if we include the perimeter length functional, or the Euler functional (number of components minus number of holes). The main question addressed by the paper is that of when the resulting point process is well-defined: geometric arguments are used to establish conditions for the point process to be stable in the sense of Ruelle

Brownian Confidence Bands on Monte Carlo Output

International audienceWhen considering a Monte Carlo estimation procedure, the path produced by successive partial estimates is often used as a guide for informal convergence diagnostics. However the confidence region associated with that path cannot be derived simplistically from the confidence interval for the estimate itself. An asymptotically correct approach can be based on the Brownian motion approximation of the path, but no exact formula for the corresponding area-minimizing confidence region is yet known. We construct proxy regions based on local time arguments and consider numerical approximations. These are then available for a more incisive assessment of the Monte Carlo procedure and thence of the estimate itself

Countable Random Sets: Uniqueness in Law and Constructiveness

The first part of this article deals with theorems on uniqueness in law for \sigma-finite and constructive countable random sets, which in contrast to the usual assumptions may have points of accumulation. We discuss and compare two approaches on uniqueness theorems: First, the study of generators for \sigma-fields used in this context and, secondly, the analysis of hitting functions. The last section of this paper deals with the notion of constructiveness. We will prove a measurable selection theorem and a decomposition theorem for constructive countable random sets, and study constructive countable random sets with independent increments.Comment: Published in Journal of Theoretical Probability (http://www.springerlink.com/content/0894-9840/). The final publication is available at http://www.springerlink.co

R-local Delaunay inhibition model

Let us consider the local specification system of Gibbs point process with inhib ition pairwise interaction acting on some Delaunay subgraph specifically not con taining the edges of Delaunay triangles with circumscribed circle of radius grea ter than some fixed positive real value $R$. Even if we think that there exists at least a stationary Gibbs state associated to such system, we do not know yet how to prove it mainly due to some uncontrolled "negative" contribution in the expression of the local energy needed to insert any number of points in some large enough empty region of the space. This is solved by introducing some subgraph, called the $R$-local Delaunay graph, which is a slight but tailored modification of the previous one. This kind of model does not inherit the local stability property but satisfies s ome new extension called $R$-local stability. This weakened property combined with the local property provides the existence o f Gibbs state.Comment: soumis \`{a} Journal of Statistical Physics 27 page

Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

A preliminary appeared as INRIA RR-5093, January 2004.International audienceIn medical image analysis and high level computer vision, there is an intensive use of geometric features like orientations, lines, and geometric transformations ranging from simple ones (orientations, lines, rigid body or affine transformations, etc.) to very complex ones like curves, surfaces, or general diffeomorphic transformations. The measurement of such geometric primitives is generally noisy in real applications and we need to use statistics either to reduce the uncertainty (estimation), to compare observations, or to test hypotheses. Unfortunately, even simple geometric primitives often belong to manifolds that are not vector spaces. In previous works [1, 2], we investigated invariance requirements to build some statistical tools on transformation groups and homogeneous manifolds that avoids paradoxes. In this paper, we consider finite dimensional manifolds with a Riemannian metric as the basic structure. Based on this metric, we develop the notions of mean value and covariance matrix of a random element, normal law, Mahalanobis distance and X² law. We provide a new proof of the characterization of Riemannian centers of mass and an original gradient descent algorithm to efficiently compute them. The notion of Normal law we propose is based on the maximization of the entropy knowing the mean and covariance of the distribution. The resulting family of pdfs spans the whole range from uniform (on compact manifolds) to the point mass distribution. Moreover, we were able to provide tractable approximations (with their limits) for small variances which show that we can effectively implement and work with these definitions

Quantum Mechanics from Focusing and Symmetry

A foundation of quantum mechanics based on the concepts of focusing and symmetry is proposed. Focusing is connected to c-variables - inaccessible conceptually derived variables; several examples of such variables are given. The focus is then on a maximal accessible parameter, a function of the common c-variable. Symmetry is introduced via a group acting on the c-variable. From this, the Hilbert space is constructed and state vectors and operators are given a clear interpretation. The Born formula is proved from weak assumptions, and from this the usual rules of quantum mechanics are derived. Several paradoxes and other issues of quantum theory are discussed.Comment: 26 page

Stochastic thermodynamics of holonomic systems

International audienc