Gibbs point processes for studying the development of spatial-temporal stochastic processes

2001 ◽  
Vol 36 (1) ◽  
pp. 85-105 ◽  
Author(s):  
Eric Renshaw ◽  
Aila Särkkä
Keyword(s):  
Stochastic Processes ◽  
Point Processes ◽  
Gibbs Point Processes

scholarly journals Existence of Gibbs point processes with stable infinite range interaction

2020 ◽  
Vol 57 (3) ◽  
pp. 775-791
Author(s):  
David Dereudre ◽  
Thibaut Vasseur
Keyword(s):  
Point Processes ◽  
Existence Results ◽  
Pairwise Interaction ◽  
Range Interaction ◽  
Pairwise Interactions ◽  
Gibbs Point Processes ◽  
Infinite Range Interactions ◽  
Finite Configuration ◽  
Multi Body ◽  
Infinite Range

AbstractWe provide a new proof of the existence of Gibbs point processes with infinite range interactions, based on the compactness of entropy levels. Our main existence theorem holds under two assumptions. The first one is the standard stability assumption, which means that the energy of any finite configuration is superlinear with respect to the number of points. The second assumption is the so-called intensity regularity, which controls the long range of the interaction via the intensity of the process. This assumption is new and introduced here since it is well adapted to the entropy approach. As a corollary of our main result we improve the existence results by Ruelle (1970) for pairwise interactions by relaxing the superstabilty assumption. Note that our setting is not reduced to pairwise interaction and can contain infinite-range multi-body counterparts.

scholarly journals Fast approximation of the intensity of Gibbs point processes

2012 ◽  
Vol 6 (0) ◽  
pp. 1155-1169 ◽  
Author(s):  
Adrian Baddeley ◽  
Gopalan Nair
Keyword(s):  
Point Processes ◽  
Gibbs Point Processes

scholarly journals Extended and conditional versions of the PASTA property

1990 ◽  
Vol 22 (2) ◽  
pp. 510-512 ◽  
Author(s):  
Dieter König ◽  
Volker Schmidt
Keyword(s):  
Stochastic Processes ◽  
Point Processes ◽  
Long Run ◽  
Run Time

Two types of conditions are discussed ensuring the equality between long-run time fractions and long-run event fractions of stochastic processes with embedded point processes. Modifications of this equality statement are considered.

A class of interacting marked point processes: rate of convergence to equilibrium

2002 ◽  
Vol 39 (1) ◽  
pp. 137-160 ◽  
Author(s):  
G. L. Torrisi
Keyword(s):  
Stochastic Processes ◽  
Rate Of Convergence ◽  
Point Processes ◽  
Marked Point ◽  
Marked Point Processes ◽  
Convergence To Equilibrium ◽  
Birth And Death Processes ◽  
Loss Networks

In this paper we obtain the rate of convergence to equilibrium of a class of interacting marked point processes, introduced by Kerstan, in two different situations. Indeed, we prove the exponential and subexponential ergodicity of such a class of stochastic processes. Our results are an extension of the corresponding results of Brémaud, Nappo and Torrisi. The generality of the dynamics which we take into account allows the application to the so-called loss networks, and multivariate birth and death processes.

Modelling Tree Roots in Mixed Forest Stands by Inhomogeneous Marked Gibbs Point Processes

2009 ◽  
Vol 51 (3) ◽  
pp. 522-539 ◽  
Author(s):  
Stefanie Eckel ◽  
Frank Fleischer ◽  
Pavel Grabarnik ◽  
Marian Kazda ◽  
Aila Särkkä ◽  
...  
Keyword(s):  
Point Processes ◽  
Mixed Forest ◽  
Forest Stands ◽  
Tree Roots ◽  
Gibbs Point Processes

scholarly journals Multifractal Analysis of Infinite Products of Stationary Jump Processes

2010 ◽  
Vol 2010 ◽  
pp. 1-26
Author(s):  
Petteri Mannersalo ◽  
Ilkka Norros ◽  
Rudolf H. Riedi
Keyword(s):  
Stochastic Processes ◽  
Multifractal Analysis ◽  
Point Processes ◽  
Multifractal Spectrum ◽  
Jump Processes ◽  
Poisson Point Processes ◽  
Infinite Products ◽  
Stationary Measures ◽  
Basic Properties ◽  
Side Result

There has been a growing interest in constructing stationary measures with known multifractal properties. In an earlier paper, the authors introduced themultifractal products of stochastic processes(MPSP) and provided basic properties concerning convergence, nondegeneracy, and scaling of moments. This paper considers a subclass of MPSP which is determined by jump processes with i.i.d. exponentially distributed interjump times. Particularly, the information dimension and a multifractal spectrum of the MPSP are computed. As a side result it is shown that the random partitions imprinted naturally by a family of Poisson point processes are sufficient to determine the spectrum in this case.

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