Fakultät für Mathematik und Naturwissenschaften

Mathematisches Kolloquium

30. November 2010, 16.15 Uhr, Hörsaal wird noch bekanntgegeben

Prof. Dr. Peter Kloeden (Johann Wolfgang Goethe Universität Frankfurt am Main)
Random attractors and the preservation of synchronization in the presence of noise

The long term behaviour of dissipatively synchronized deterministic systems is determined by the system with the averaged vector field of the original uncoupled systems. This effect is preserved in the presence of environmental i.e., background or additive noise provided stochastic stationary solutions are used instead of steady state solutions. Random dynamical systems and random attractors provide the appropriate mathematical framework for such problems and require Ito stochastic differential equations to be transformed into pathwise random ordinary differential equations. An application to a system of semi-linear parabolic stochastic partial differential equations with additive space-time noise on the union of thin bounded tubular domains separated by a permeable membrane will be considered. What happens with linear multiplicative noise will also be considered.

This a joint work with Tomas Caraballo (Sevilla) and Igor Chueshov (Kharkov). Based on the papers

  • T. Caraballo and P.E. Kloeden, The persistence synchronization under environmental noise Proc. Roy. Soc. London. A 461 (2005), 2257-2267.
  • T. Caraballo, I. Chueshov and P.E. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal. 38 (2007), 1489-1507.

2. November 2010, 18.30 Uhr, Hörsaal 26 (Raum I.13.65)

Prof. Dr. Bernd Kawohl (Universität zu Köln)

Gleichdicke, oder warum konvexe Geometrie Leben retten kann

Wann hat ein Rohr einen exakt kreisförmigen Querschnitt?
Wenn es aus jeder Richtung von außen gleich dick zu sein scheint?

Das könnte man mit einer Schieblehre nachmessen, und so gaben es die Vorschriften der NASA beim Raketenbau vor. Den Verfassern der Vorschriften war entgangen, dass es geometrische Formen gibt, so genannte Gleichdicke, die nicht Kreise sind. Letztlich führte dieser Irrtum zum Tod einer ganzen Raumschiffbesatzung. Im Vortrag wird dargelegt, wo uns Gleichdicke im täglichen Leben überall begegnen und welche interessanten mathematischen Fragen mit ihnen verbunden sind.

Der Vortrag richtet sich an die mathematisch interessierte Öffentlichkeit und nicht nur an Mathematikstudenten.

7. April 2010, 14.15 Uhr, Hörsaal 6 (Raum G.10.06)

Dr. Jörg Kampen (WIAS Berlin)

On a class of semi-elliptic diffusion models.
Part I: a constructive analytical approach for global existence, densities, and numerical schemes (with applications to the Libor market model)

Computational parsimony makes reduced factor Libor market models popular among practioners. However, value functions and sensitivities of such models are described by degenerate parabolic (i.e. semielliptic) equations where the existence of regular global solutions is not trivial. In this talk, we show that for a considerable class of degenerate equations (including equations corresponding to reduced LIBOR market models of practical interest) regular global solutions can be constructed.

The result is also of interest for the theory of degenerate parabolic equations. In addition, the constructive proof of the global existence result allows to derive explicit approximations for the transition probabilities. These transition probabilities then lead to sophisticated Monte-Carlo schemes for semielliptic diffusion models (subsuming projective Markovian models). Moreover, recent results on bounded variance estimators for Greeks of valuations under such schemes are generalized to reduced factor models. The emphasis in the present part of our treatment of reduced factor models is on conceptual and (constructive) analytical issues. A more detailed analysis of numerical and computational issues, as well as quantitative experiments will be content of the second part.

(joint work with Christian Fries)

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