Research Program

The collobration of the International Graduate School will be focused on:

(1) Dirichlet forms and stochastic partial differential equations: Z. M. Ma (Beijing), M. Roeckner(Bielefeld)

On the theoretical side, the theory of generalized Dirichlet forms is presently being developed, directed towards new applications. A number of challenging problems of fundamental importance, e.g., in the theory of Markov processes, are in the centre of interest. Generalized Dirichlet form theory also gives rise to a new method to construct and analyze weak solutions to stochastic partial differential equations (SPDE) with singular coefficients.

On the applied side, particular classes of SPDE, e.g., in hydrodynamics such as the stochastic generalized Burgers or Navier-Stokes equations, are in the focus of studies. Furthermore, backward stochastic differential equations along the lines of recent work of Bally, Fuhrman, Pardoux, Stoica, Tessitore and others are being studied, in particular with respect to their applications to stochastic finance.

(2) Gaussian analysis and applications: S. L. Luo (Beijing), J. A. Yan (Beijing), L. Streit (Bielefeld)

One main point of investigation here are spaces of distributions (generalized functions) in infinitely many variables in the framework of generalized properly weighted Wiener chaos type decompositions (i.e., the white noise or Hida calculus). This has led to powerful generalizations of the Clark-Ocone formula and the construction of self-intersection local times of Brownian motion in arbitrary dimensions. Generalizations and extensions thereof are being pursued presently.

The application of Gaussian analysis in finance is one part of the work. Furthermore, Feynman path integrals and applications to quantum field theory are in the centre of considerations. Functional integrals have become more and more important not only in mathematical physics, but also in areas within pure mathematics such as topology or geometry (Atiyah, Bismut, Konsevich, Witten and many others). Recently, Léandre has developed a new variant of Hida calculus specially tailored for applications in index theory and non-commutative geometry. Present day research also concentrates (among other things) on extending recent applications of chaos decomposition techniques to certain stochastic (ordinary and partial) differential equations, in particular those with a fractional Brownian motion in the noise part.

(3) Stochastics analysis on configuration spaces: M. F. Chen (Beijing), Y. Kondratiev (Bielefeld)

The theoretical part of this field is concerned with the precise mathematical analysis of configuration spaces of particles in continuum (e.g. in R^d or a manifold), especially, with their various geometrical structures, suitable metrics, and their harmonic analysis. Partly, this is strongly related to the theory of Dirichlet forms, since the latter reflect the geometry of the underlying space and give rise to the construction of stochastic dynamics of the particles even when interactions are realistic, hence very singular. Applications have a wide range, including interacting infinite particle systems in statistical mechanics, but also in economics. One particular recent development should be emphasized, namely the rigorous construction of birth and death processes in continuum in infinite volume. Furthermore, a stochastic dynamics describing a system of infinite interacting particles of pure jump type exhibiting completely new features as a conservative dynamic with respect to their spectral or scaling properties.

(4) Complex systems, random graphs and networks: F. Z. Gong (Beijing), Ph. Blanchard (Bielefeld)

In this area a distinction of the research parts in more theoretical or more applied aspects is impossible, since there is no sufficiently developed probabilistic theory for this kind of complex systems yet. Random graphs and networks appear everywhere in real life situations (world wide web, social networks, biochemical networks, etc.) and there is an enormous activity in analyzing and trying to understand them. Mathematically rigorous results are comparably rare in regard to the overall development in the field. This exactly makes the area attractive and prospectively fertile for mathematicians, in particular probabilists.

(5) Transportation cost, functional and isoperimetric inequalities: F. Y. Wang (Beijing), F. Gotze (Bielefeld)

This area is devoted to functional and related (transportation cost, isoperimetric) inequalities. The interplay between the probabilistic coupling methods, its consequences for such inequalities, and the related spectral information led to a flourishing area of its own on this interface between stochastics and analysis. Fascinating problems as to what extent the famous Talagrand inequality for the standard Gaussian measure holds for infinite dimensional manifolds such as paths and loop spaces over Riemannian manifolds (without group structure) are still open. Also weak type Talagrand inequalities, analogues of the well-known weak Poincare inequalities, are still to be found and to be investigated. But even on the technically much simpler level of Markov chains many questions in connection with transportation cost inequalities and couplings are open and of high interest. Applications here are directed towards obtaining spectral information for infinite dimensional stochastic systems, such as infinite particle systems both on the lattice and in the continuum, and both in mathematical physics and economics.

(6) Stochastics, economics and finance: J. Xia (Beijing), J. A. Yan (Beijing), H. Dawid (Bielefeld)

This area is, of course, very wide. One topic is, e.g., the application of dynamic stochastic methods in economics with particular focus on allocation problems in teams. One interesting theoretical question is in how far existing insights about the limit behavior ("stochastically stable states") of perturbed stochastic game dynamics can be extended to provide characterizations in more general settings, in particular concerning the transient behavior of such processes which is typically ignored in the literature. The applications lie in the area of mechanism design and implementation. Another application is in the mathematical finance field dealing with the question under which circumstances allocation rules and cooperative trading strategies within teams of investors exist which are efficient and truth revealing with respect to the investors preferences. Another application is about designing mechanisms for hold-up problems in team production with two-sided (non-contractible) specific investments such that states with efficient investment behavior are stochastically stable.

Another topic concerns models in stochastic finance in which questions such as utility optimization or risk minimization in the context of risk measures or securitization concepts are formulated in the framework of stochastic control theory and PDE in the viscosity sense, their interpretation as nonlinear conditional expectations, and their application in models of financial risk is in the focus of interest of various research groups in China an Germany.