98 research outputs found
Dynamics of spatial logistic model: finite systems
The spatial logistic model is a system of point entities (particles) in
which reproduce themselves at distant points (dispersal) and
die, also due to competition. The states of such systems are probability
measures on the space of all locally finite particle configurations. In this
paper, we obtain the evolution of states of `finite systems', that is, in the
case where the initial state is supported on the subset of the configuration
space consisting of finite configurations. The evolution is obtained as the
global solution of the corresponding Fokker-Planck equation in the space of
measures supported on the set of finite configurations. We also prove that this
evolution preserves the existence of exponential moments and the absolute
continuity with respect to the Lebesgue-Poisson measure.Comment: To appear in "Semigroups of Operators: Theory and Applications.
Bedlewo 2013" Springer Proceedings in Mathematic
Self-regulation in the Bolker-Pacala model
The Markov dynamics is studied of an infinite system of point entities placed
in \mathds{R}^d, in which the constituents disperse and die, also due to
competition. Assuming that the dispersal and competition kernels are continuous
and integrable we show that the evolution of states of this model preserves
their sub-Poissonicity, and hence the local self-regulation (suppression of
clustering) takes place. Upper bounds for the correlation functions of all
orders are also obtained for both long and short dispersals, and for all values
of the intrinsic mortality rate.Comment: arXiv admin note: substantial text overlap with arXiv:1702.0292
Gibbs random fields with unbounded spins on unbounded degree graphs
Gibbs random fields corresponding to systems of real-valued spins (e.g.
systems of interacting anharmonic oscillators) indexed by the vertices of
unbounded degree graphs with a certain summability property are constructed. It
is proven that the set of tempered Gibbs random fields is non-void and weakly
compact, and that they obey uniform exponential integrability estimates. In the
second part of the paper, a class of graphs is described in which the mentioned
summability is obtained as a consequence of a property, by virtue of which
vertices of large degree are located at large distances from each other. The
latter is a stronger version of a metric property, introduced in [Bassalygo, L.
A. and Dobrushin, R. L. (1986). \textrm{Uniqueness of a Gibbs field with a
random potential--an elementary approach.}\textit{Theory Probab. Appl.} {\bf
31} 572--589]
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