Stochastic Analysis

Abstract

The meeting took place on June 2–6, 2008, with over 45 people in attendance. Each day had 5-6 talks, except for Wednesday: the traditional hike took place in the afternoon, despite somewhat gray skies.

The talks reviewed several directions in which progress in the general field of stochastic analysis occurred since the last meeting of this theme in Oberwolfach three years ago. As has become a tradition, several themes were covered in some depth, in addition to a broad overview of recent development. Among these themes a prominent role was played by the SLE (Schramm–Loewner equation), that appeared in talks on scaling scaling limits of dynamical percolation, Ising models on (a wide class of) graphs, and scaling limits of triangulations. Two other main themes were also present: the zeroes of random analytic functions (including the eigenvalues of random matrices), and the long time behavior of stochastic flows motivated by fluid dynamics. What follows is a brief description of the topics covered in the talks.

A cluster of talks discussed the zero set of random functions (including the eigenvalues of random matrices). In the first lecture of the week, B. Tsirelson discussed moderate deviations for random fields, motivated by the asymptotic normality of the process of zeroes of random analytic functions; the result presented is not restricted however to this application. M. Sodin reported on his work with F. Nazarov, where they studied the number of nodal domains for random spherical harmonics. Concentration played an important role, and many exciting open questions were discussed. Later in the week, M. Krishnapur described the determinantal process arising from considering the zeroes of the determinant of random analytic matrices, and managed to provide a full proof within 30 minutes! Following his work with B. Valko, B. Virag introduced the “Brownian caroussel”, a diffusion process closely related to the study of eigenvalues in the bulk of the spectrum for the $\beta$-ensembles of (tri-diagonal) random matrices, and showed how asymptotics of gap probabilities can be computed using this process. N. O'Connell explained how a diffusion process studied by Matsumoto and Yor was used (in his work with Baudoin) to study the partition function of polymer measures.

Another cluster of talks discussed various scaling limits of processes in the plane. O. Schramm and C. Garban described their work (joint with G. Pete) concerning the scaling limit of dynamical percolation, which exhibits several nice properties, such as conformal invariance and Markovianity. Y. LeJan, motivated by Lawler–Werner's Loop Soup, discussed first on a base graph and then in an infinite dimensional situation (where a renormalization procedure is employed) measures on ensembles of loops, with Dynkin's isomorphism theorem and certain “$\alpha$-permanents” playing a role. A cluster of 3 talks discussed the Ising model over finite (non periodic) graphs. C. Boutillier reviewed the notion of isoradial graphs and the associated statistical mechanics of dimer configurations on decorated graphs, and discussed a formula derived by him and B. de Tiliere for the free energy of the Ising model. D. Chelkak and S. Smirnov then described, in back to back talks, their work on convergence of harmonic measures, Poisson kernels, and observables, for the Ising model, when the mesh size goes to 0, to the SLE based limits, when no special periodic structure of the mesh approximation is assumed. This is the first time where conformal invariance of a model from statistical physics for a large class of graphs is derived. The talk of V. Beffara described conditions on sequences of iterations of a triangulation that lead to site percolation probabilities converging to conformally invariant limits. Finally, N. Makarov discussed conformal field theory from a complex analyst point of view, and D. Wilson described his joint work with Rick Kenyon deriving formulas for the probabilities of the different possible node connections in a grove (a grove is a planar spanning forest in which every component tree contains one or more of a specified set of vertices on the outer face) or dimer models. These results are related to multiple SLE crossing probabilities.

The last thematic cluster of talks related to infinite dimensional flows. M. Hairer described a condition (based on the strong Feller property), that ensures ergodicity of flows by a coupling approach. This was followed by J. Mattingly, who related infinite dimensional ergodicity to the study of the Navier–Stokes equation. S. Kuksin's talk dealt with 2D turbulence in the Eulerian limit.

The other talks were on other directions but together gave a broad view of current progress in stochastic analysis. E. Maurel-Segala described the combinatorics of matrix models and the rigorous evaluation of terms in the expansion of the free energy. S. Chatterjee described a new, Stein-method based, proof of the strong embedding of random walk into a Brownian motion (a result originally due to Komlos–Major–Tusjnadi). G. Ben Arous identified all possible limits of randomly trapped random walks, as appearing for example in the context of random walk on the incipient critical Galton–Watson tree. Anton Bovier explained us how ageing takes place at intermediate time scales for spin glass models, in particular showing the connection with the universality of random traps as in Ben Arous talk. T. Kumagai described joint work with M. Barlow in which devoted to construction and uniqueness of Brownian motions on the Sierpinski carpet. M. von Renesse presented joint work with K-T. Sturm where they construct a natural “canonical process” on the set of probability distributions on the unit circle, where the space of measures is equipped with the Wasserstein distance. A. Dembo in a joint work with A. Montanari described Ising models on locally tree-like graphs, in particular he gave a very explicit method in order to compute the free entropy density in this context. O. Angel discussed the speed process of various types of multi-class totally asymmetric exclusion process. Den Hollander explained a quenched large deviation result (joint work with Birkner and Greven) for the empirical process of words cut out from a letter sequence via a renewal process. Finally Terry Lyons talked about the signature of a path: this is a map from a path in ${\mathbb{R}}^d$ to the infinite vector of its iterated integrals/ Levy areas. The algebraic and topological properties of this map were discussed as well as applications for numerical computation of expected values of SDE.