Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Oleksandr Kutovyi
Bielefeld Universität
Markov Jump Dynamics in Continuum: State Evolution and Vlasov Scaling
We investigate a stochastic (conservative) jump dynamics of interacting particles in continuum. We
consider objects dual to correlation functions (respectively correlation measures), which are called quasiobservables.
By solving the evolution equation for quasi-observables, we obtain the non-equilibrium
evolution of these objects. The corresponding dual equation describes the time evolution of correlation
functions and generalizes the BBGKY-hierarchy from Hamiltonian to jump dynamics. By means of the
duality between quasi-observables and correlation functions we can transfer the evolution of quasiobservables
to correlation functions. Afterwards we perform a Vlasov-type scaling of the dynamics, which
leads to a rescaled and limiting evolution of correlation functions. We show convergence of the solution of
the rescaled hierarchy to the solution of the limiting one. The chaos preservation property of the limiting
hierarchy leads to the derivation of a kinetic equation for the particle density which is a Vlasov-type
equation for the considered model.
Eugene Lytvynov
Swansea University
Determinantal point processes with J-Hermitian correlation kernels
Let X be a locally compact Polish space. The configuration space over X, denoted by Γ, is defined as the
set of all locally finite subsets of X. A probability measure on Γ is called a point process. A point process
µ can be described with the help of correlation functions: the n-th correlation function is a non-negative
symmetric function k (n) (x1,...,xn) on X n which may be heuristically interpreted as the µ-probability to
find points x1,...,xn in a configuration. A point process is called determinantal if there exists a function
K(x,y) on X 2 , called the correlation kernel, such that the n-th correlation function of this point process
is given by det[K(xi,xj)] n i, j=1 . If the correlation kernel is Hermitian, the Macchi–Soshnikov theorem gives
a necessary and sufficient condition for the existence of a determinantal point process with correlation
kernel K(x,y). We will derive a counterpart of this theorem in the case where the correlation kernel is
J-Hermitian, i.e., Hermitian in an indefinite scalar product. Such point processes occurred in Borodin and
Olshanski’s studies on harmonic analysis of the infinite symmetric group.
Maria João Oliveira
Universidade Aberta/CMAF
Glauber dynamics in the continuum via generating functionals evolution
We construct the time evolution of Glauber dynamics for a continuous infinite particle system in terms of
generating functionals. This is carried out by Peano-type approximations in a scale of Banach spaces,
leading to a local (in time) solution which may be extended to a global one. An application of this approach
to Vlasov-type scaling in generating functionals is considered as well.
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