We set out below a snapshot overview of the direction in which the teams expect the research to go, it is based on a plan which was requested by the European Commission, and drawn up by the teams while meeting together in Barcelona in the summer of 1996. We include it here to help anyone considering applying to get a picture of research interests.
It is the nature of good research that it is full of surprises, and any such overview should be treated with caution. Candidates feeling they could work with the teams should not in any way be put off applying because their research does not seem to be mentioned below.
Stochastic Analysis has emerged as a core area of mathematics interacting with other central areas of mathematics and science. In this project some of the more active teams in stochastic analysis will join their efforts to develop a coherent technology for studying random phenomena and dynamical systems.
The central goal will be to unify concepts and techniques for the analysis of high-dimensional stochastic phenomena which are modelled by measures on infinite-dimensional spaces. This search will be carried out by working in depth on concrete path and configuration spaces which are motivated by their relevance to concrete problems, (e.g. in physics and finance), where such measures arise naturally. Basic examples include path and configuration spaces on manifolds or trees and on spaces of measures and of smooth transformations. The measures are often specified by a Markovian dynamics or by a spatial Gibbs structure. The stochastic analysis of such models will involve techniques of infinite-dimensional calculus which use the structure of the underlying measure. They are centered centered around the key tools of Itô calculus, semigroup analysis and large deviations. We hope common structures and new tools will emerge from these concrete case studies.
One key approach is to work directly in the infinite-dimensional setting. A number of analytical problems given on a finite-dimensional state space can be lifted to the infinite-dimensional stochastic setting, can be clarified and solved there, and the solution can be successfully transferred back to the finite level. This is illustrated by the Malliavin approach to regularity properties of partial differential equations. In the same spirit, we will explore the connections between super-processes and singular boundary value problems for some nonlinear partial differential equations, and the evolution and equilibrium properties of particle systems and random media.
We hope thst mathematical results in the infinite-dimensional context will provide qualitative and quantitative insight into the behaviour of large finite stochastic systems in the same way that the thermodynamic formalism of statistical mechanics provides an illuminating infinite-dimensional approximation to critical phenomena in finite systems. One major thrust of our project is to turn such analogies into rigorous mathematical results. This joint effort will involve a combination of techniques from theory of Dirichlet forms, large deviations and statistical mechanics. A key feature of this project is that path and configuration spaces will be viewed as infinite dimensional analogues of finite dimensional manifolds. Thus, concepts and methods of topology, geometry and analysis will emerge as new tools in the infinite-dimensional stochastic setting.
This progress in core areas of infinite-dimensional stochastic analysis is expected to have an impact in other fields. Some of the participants will be actively involved in promoting such applications, including the connections to quantum field theory, statistical mechanics, control theory and mathematical finance.
To pursue the central goal of expanding the scope and the power of the methods of stochastic analysis, the participants have identified six intertwined and vital problem areas where several teams will join their efforts and where progress would lead to essentially better understanding of stochastic analysis. These areas are structured so as to bring our expertise in stochastic analysis together for this project.
The teams believe the questions are important, but they would want to emphasise that they are by no means the only questions (or sometimes even the main questions) in those areas.
Stochastic evolution equations are one of the most basic and classical parts of stochastic analysis, providing historic motivation for the development of stochastic calculus as the most powerful tool in Stochastic Analysis today. Describing and approximating the path properties of solutions to stochastic evolution equations and their backward counterparts remains the central problem and this will be addressed by the participants.
The study of stochastic dynamics for particle systems present challenging problems for the participant teams in IMPCOL, Bielefeld, Berlin and Paris particularly in terms of phase transitions and ergodicity.
Stochastic flows are a special class of evolution equations with values in spaces of diffeomorphisms. The participants in Barcelona, IMPCOL and Orsay will investigate their dynamical and geometric properties exploiting the extra geometric structure of these systems to pull a number of threads of the project together.
The parallels between the finite and infinite dimensional setting can be seen most clearly when directly studying the geometry of the key infinite dimensional spaces. Many ideas can be modified to go across---but almost always the transfer of these ideas is instructive and not at all automatic. The participants will particularly seek to identify geometric and analytic properties of path spaces, aiming to establish a foundation for a coherent geometric analysis of such spaces.
There are interactions with topology which will be particularly brought out in Bielefeld as they try to use stochastic analyse tools to construct topological invariants from random fieldsdevelop topological invariants using random fields.
One significant benchmark would be for the participants to establish appropriate log-Sobolev inequalities for Wiener measure on Loop spaces.
The coding of Axiom A flows has made a strong connection between Ergodic theory, statistical mechanics, and stochastic methods and greatly improves understanding of the area. Statistical and Ergodic properties of disssipative non-uniformly hyperbolic dynamical systems are a focus of participant attention in Stockholm. Stochastic diffusions on foliations provide another bond relating stochastic analysis to dynamical systems and will be used in Orsay to develop specific asymptotics for action integrals. Stochastical dynamical systems arising in quantum theory will also be studied.
The participants will study and further develop the emerging theory of disordered systems with regard to both dynamical and equilibrium properties including bubble formation and condensation.
Particular attention will be paid to exploiting the Stochastic Analysis tools of large deviations and hydrodynamic limits. The participant teams will study aggregation and growth models, often involving the detailed analysis of phase transitions. In IMPCOL, Stockholm, and Berlin some emphasis will be given to shape formation such as in Diffusion Limited Aggregation (DLA).
Super-processes are scaling limits of branching Markov processes and form a relatively new and central class of infinite dimensional processes. Their importance is in part because they solve certain nonlinear stochastic partial differential equations. This leads to qualitative properties for the solutions to SPDEs.
Super-processes already provide an effective tool to analyse certain semi-linear pdes. In this context, a central objective of Paris, Berlin and IMPCOL is to arrive at a good understanding of the relationship between super-processes, stochastic partial differential equations and interacting particle systems.
A related problem to be addressed by the participants in Paris and Orsay is to extend the probabilistic approach, and particularly the snake method (the snake process is a special path valued Markov process) to equations with a non-quadratic non-linear term.
This area has developed very vigorously and is now one of the most active fields of application for the concepts and techniques of stochastic analysis. One major focus will be the stochastic analysis of incomplete financial markets. The teams in Berlin and Paris plan to clarify the relation between different martingale measures, different optimality criteria and different restrictions on hedging strategies, the team in Toulouse will explore the connections to non-linear partial differential equations and to backward stochastic differential equations. The overall purpose is to integrate these different approaches and to develop a coherent and flexible methodology for hedging the risks due to incompleteness. This would be a major theoretical advance in the field.
A second major goal is to understand the structure of martingale measures in the infinite-dimensional context of stochastic models for the term structure of interest rates. A key problem is the construction of approximations that reflect the very high dimensional system and yet induce efficient low dimensional choices of the financial instruments used in dynamic strategies for hedging derivatives of the term structure. Here the expertise of those teams which focus on infinite-dimensional methods and on their finite-dimensional approximation (Bielefeld, IMPCOL, Paris) will be crucial.
The progress of the project will undoubtedly lead to an increasing range of applications.
IMPCOL intends to research the application of the measure valued processes in order to construct good finite dimensional approximations to the intrinsically infinite dimensional solution of the non-linear filtering equation.
Toulouse will research applications of log-Sobolev inequalities to random algorithms, especially in order to obtain explicit and tractable bounds on rates of convergence.
One goal of the studies in interacting particles is to create a better understanding of stochastic networks such as those arising in communication systems. This is another example of an important class of applications which both uses and influences theoretical applications.
GFM
<gfm@alf1.cii.fc.ul.pt>,
JMCE
<jmce@titania.cii.fc.ul.pt>
URL:
http://alf3.cii.fc.ul.pt/gfm/Projects/SA_TMR_outline.en.html
Last modified: Mon Feb 3 21:12:11 WET 1997