Stochastic Analysis and Stochastic Partial Differential Equations (12w5023)
Robert Dalang (Ecole Polytechnique Fédérale de Lausanne)
Davar Khoshnevisan (University of Utah)
Yimin Xiao (Michigan State University)
The past ten or so years has witnessed the discovery of a number of deep connections between stochastic PDEs and the theories of Gaussian processes, infinite-dimensional Markov processes, probabilistic potential theory, and mathematical physics.
Typically, solutions to linear SPDEs driven by Gaussian noise are Gaussian random fields, or at least, generalized Gaussian random fields. Therefore, techniques from the theory of Gaussian processes can be used to address questions of sample path continuity and Holder continuity. This gives an indication of what sort of result can be expected for nonlinear SPDEs, and it is often a substantial challenge to establish the corresponding result for nonlinear forms of the SPDE.
For instance, solutions to the linear stochastic heat or wave equation driven by spatially homogeneous noise is a generalized Gaussian random field, and one can determine necessary and sufficient conditions for the solution to be an ordinary random field. It is a challenge to show that under the same conditions, the nonlinear stochastic heat or wave equations have a random-field solution. This has been addressed in works of Dalang, and of Peszat and Zabczyk, Nualart and Viens, and Foondun and Khoshnevisan. This last reference also contains an unexpected connection with existence of local times of Markov processes.
A second set of issues concerns Holder continuity. Again, for a linear SPDE, older continuity properties of the solution can in principle be determined by methods from the theory of anisotropic Gaussian processes, as developed recently by Bierm'e, Lacaux and Xiao. This again indicates what are likely to be the best possible results for nonlinear SPDEs. Establishing such Holder continuity results may be quite a challenge, as occurs in the AMS Memoir of Dalang and Sanz-Sol'e.
A third example of the type of connection that we are interested in comes from probabilistic potential theory. For Markov processes, potential theory is a mature subject, and for Gaussian random fields, numerous results are available, many of which can be found in the 2002 monograph of Khoshnevisan. The behavior of the Gaussian random field is determined by its covariance function, and estimates on this function are needed. These yield results on hitting properties for solutions to linear SPDEs, such as the stochastic heat and wave equations in various spatial dimensions (see the works of Khoshnevisan and Shi, Bierme, Lacaux and Xiao, and others). Several extensions of these results to nonlinear SPDEs have been obtained recently by Dalang, Khoshnevisan, Nualart, and Sanz-Sol'e. These extensions make extensive use of Malliavin's calculus in order to establish heat-kernel-type bounds for various probability density-functions connected to the solution of the SPDE. Those bounds, in turn, replace covariance-functionestimates in the Gaussian case.
We anticipate that there are many other connections to explore, concerning properties of level sets, local times, large deviations, small ball properties, etc. This meeting will be the occasion to identify such connections and to develop appropriate research directions.
Within the theory of SPDEs, many questions are motivated by mathematical physics, and, in particular, by the study of intermittency, complexity, and turbulence. SPDE models include the classical Navier--Stokes equations perturbed by noise (Mikulevicius, Rozovskii, Flandoli, Mattingly, Hairer, dots), stochastic Euler equations (Brzezniak, Peszat), models specific to certain applications, in particular for climatology and geophysics (Majda, Franzke, Khouider), and the mathematical structure of kinetically-grown random surfaces (Carmona, Molchanov, Tindel, Viens, and others). Important open questions include the issue of regularization by noise (Cerrai, Freidlin), avoidance of singularities by the addition of noise (Da Prato, Debussche), viscosity solutions (Souganidis)-- which are ingredients in the study of well-posedness of PDEs and SPDEs --including the millennium problem about well-posedness of the 3D Navier-Stokes equations, and the sizes of the ``islands'' in SPDEs with intermittent solutions. An objective of the meeting is to bring together experts in the field of SPDEs and turbulence to identify the state of the art as well as important open problems for future research.
The interplay between Gaussian random field, SPDEs and mathematical physics has already motivated a substantial body of research. The proposers believe that these connections point to new and interesting directions of research. The proposed meeting is expected to help accelerate the progress in these central subjects.