Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Tobias Kuna
University of Reading
Truncated moment problem and Realizability of point processes
To reconstruct in a systematical way from observable quantities, the underlying effective description of a
complex system on relevant scales is a task of enormous practical relevance. Realizability considers the
partial question if the system can be described by point-like objects on the relevant scale, cf. Percus(1964)
and Crawford et al. (2003). In this talk, based on Kuna et al. (2009), the realizability problem is introduced.
It is identified as an infinite dimensional version of the classical truncated power moment problem.
One can associate a linear functional on the space of polynomials to any kind of moment problem. A
classical theorem for complete moment sequences, see e.g. Haviland(1935/6) states that solvability of
the moment problem is equivalent to positivity of this functional. However, this is wrong in general for
truncated moment problems. A new general approach for truncated moment problems will be presented
which overcomes this difficulty. To our knowledge this approach is also new for finite dimensional
problem, however it may be more adapted for infinite dimensional problems. An extension to random
closed set is considered in Lachieze-Rey et al. (2011), where the technique is also presented in general
terms. Finally, it is shown that all known restrictions for realizability can be easily derived using this criteria.
Literatur
Percus, J. K.(1964) The pair distribution function in classical statistical mechanics. In: Frisch, H. L. , Lebowitz,
J. L. (eds.) The Equilibrium theory of classical fluids. Benjamin, New York.
Crawford, J., Torquato, S., Stillinger, F. H. (2003) Aspects of correlation function realizability.
J. Chem. Phys. 119, 7065 - 7074.
Kuna T., Lebowitz J. L., Speer, E. R. (2009) Necessary and sufficient conditions for realizability of point
processes arXiv: 0910.1710. To appear in Journal of Applied Probability
Haviland (1935) On the moment problem for distributions in more than one dimension. Amer. J. Math. 57,
562-568
Haviland (1936) On the moment problem for distributions in more than one dimension II. Amer. J. Math.
58, 164-168
Lachieze-Rey R., Molchanov I. (2011) Regularity conditions in the realizability problem in applications to
point processes and random closed sets. arXiv: 1102.1950.
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