Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Dmitri Finkelshtein
Institute of Mathematics, Kiev
Semigroup approach to birth-and-death stochastic dynamics in continuum
We consider general approach using perturbation semigroup technic for construction birth-and-death
spatial dynamics for interacting particle systems
Dennis Hagedorn
Universität Bielefeld
Gibbs distributions related to a Gamma measures over the cone of positive discrete Radon
measures
A Gamma measure Gθ , θ > 0 being a shape parameter, can be regarded as the ’free’ case of a Gibbs
measure because the involved potential is zero. Vershik, Gel’fand and Graev introduced the Gamma
measure Gθ in the context of the representation theory in 1975. It can be seen as a “marked“ Poisson
measure with an (infinite) Levy measure λθ on the marks.
We construct a Gibbs measure that corresponds to a (possibly negative,) non-symmetric potential
with infinite interaction range and the to Gθ associated "marked" Poisson measure.
A biological motivation to consider a Gamma measure is that it discribes the allocation of animals of
different sizes. Namely, we can model that there are few big animals and many small ones like plancton,
which seem to be almost continuously distributed. For this a Gamma measures yields an appropriate
distribution.
But, a (’free’) Gamma measure does not take into account interaction. The interaction yields a
distribution of the size and position of animals depending on the surrounding animals. An interesting
interacting potential has an infinite interaction range and is not translation invariant. Using the (not
necessarily positive) potential we can define via a relative energy and the DLR equation the notion of a
Gibbs measure on the cone, whose existence we prove.
Yuri Kozitsky
Uniwersytet Marii Curie-SkAlodowskiej, Lublin
Evolution of states in spatial Glauber dynamics
211