Mini-Workshop: Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-Ito Chaos Expansions and Stochastic Geometry

2013 ◽  
Vol 10 (1) ◽  
pp. 483-520
Author(s):  
Matthias Reitzner ◽  
Giovanni Peccati
Keyword(s):  
Malliavin Calculus ◽  
Stochastic Geometry ◽  
Stochastic Analysis ◽  
Point Processes ◽  
Poisson Point Processes

scholarly journals Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

2020 ◽  
pp. 1-14
Author(s):  
SHOTA OSADA
Keyword(s):  
Point Processes ◽  
Duke Math ◽  
Random Point ◽  
Determinantal Point Processes ◽  
Fredholm Determinants ◽  
Poisson Point Processes ◽  
Translation Invariant ◽  
Ergodic Properties ◽  
Tree Representations ◽  
Invariant Kernels

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

scholarly journals A generalization of Matérn hard-core processes with applications to max-stable processes

2020 ◽  
Vol 57 (4) ◽  
pp. 1298-1312
Author(s):  
Martin Dirrler ◽  
Christopher Dörr ◽  
Martin Schlather
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Hard Core ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Poisson Point Processes ◽  
Original Process ◽  
Mixed Moving Maxima

AbstractMatérn hard-core processes are classical examples for point processes obtained by dependent thinning of (marked) Poisson point processes. We present a generalization of the Matérn models which encompasses recent extensions of the original Matérn hard-core processes. It generalizes the underlying point process, the thinning rule, and the marks attached to the original process. Based on our model, we introduce processes with a clear interpretation in the context of max-stable processes. In particular, we prove that one of these processes lies in the max-domain of attraction of a mixed moving maxima process.

Convergence of extreme values of Poisson point processes at small times

Extremes ◽  
2021 ◽  
Author(s):  
Boris Buchmann ◽  
Ana Ferreira ◽  
Ross A. Maller
Keyword(s):  
Point Processes ◽  
Extreme Values ◽  
Poisson Point Processes ◽  
Small Times

Application of stochastic geometry to problems in plankton ecology

1992 ◽  
Vol 336 (1277) ◽  
pp. 225-237 ◽  
Keyword(s):  
Optical Sensors ◽  
Stochastic Geometry ◽  
Point Processes ◽  
Random Distribution ◽  
Interparticle Distance ◽  
Ecosystem Dynamics ◽  
Order Moment ◽  
Patch Structure ◽  
Covariance Functions ◽  
Plankton Ecology

The most fundamental linkages in ecosystem dynamics are trophodynamic. A trophodynamic theory requires a framework based upon inter-organism or interparticle distance, a metric important in its own right, and an essential component relating trophodynamics and the kinetic environment. It is typically assumed that interparticle distances are drawn from a random distribution, even though particles are known to be distributed in patches. Both random and patch-structure interparticle distance are analysed using the theory of stochastic geometry. Aspects of stochastic geometry - point processes and random closed sets (RCS) - useful for studying plankton ecology are presented. For point-process theory, the interparticle distances, random -distribution order statistics, transitions from random to patch structures, and second-order-moment functions are described. For RCS-theory, the volume fractions, contact distributions, and covariance functions are given. Applications of stochastic-geometry theory relate to nutrient flux among organisms, grazing, and coupling between turbulent flow and biological processes. The theory shows that particles are statistically closer than implied by the literature, substantially resolving the troublesome issues of autotroph-heterotroph nutrient exchange; that the microzone notion can be extended by RCS; that patch structure can substantially modify predator-prey encounter rates, even though the number of prey is fixed; and that interparticle distances and the RCS covariance function provide a fundamental coupling with physical processes. In addition to contributing to the understanding of plankton ecology, stochastic geometry is a potentially useful for improving the design of acoustic and optical sensors

Bayesian modeling and decision theory for non-homogeneous Poisson point processes

2020 ◽  
Vol 36 ◽  
pp. 100412
Author(s):  
Jiaxun Chen ◽  
Athanasios C. Micheas ◽  
Scott H. Holan
Keyword(s):  
Decision Theory ◽  
Bayesian Modeling ◽  
Point Processes ◽  
Poisson Point Processes

scholarly journals The Cauchy–Schwarz Divergence for Poisson Point Processes

2015 ◽  
Vol 61 (8) ◽  
pp. 4475-4485 ◽  
Author(s):  
Hung Gia Hoang ◽  
Ba-Ngu Vo ◽  
Ba-Tuong Vo ◽  
Ronald Mahler
Keyword(s):  
Point Processes ◽  
Poisson Point Processes
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