# Stochastic Analysis and Non-Classical Random Processes

### Wendelin Werner

ETH Zürich, Switzerland### Jean Dominique Deuschel

Technische Universität Berlin, Germany### Ofer Zeitouni

Weizmann Institute of Science, Rehovot, Israel

## Abstract

The meeting was organized by J. D. Deuschel (Berlin), W. Werner (Orsay), and O. Zeitouni (Minneapolis). It was broadly divided into several distinct but related themes. The talks varied in length. There were two 70 minute overview talks, 12 talks of 45 minutes, and the rest were shorter 35 minutes talks.

The morning of Monday was devoted to an exposition of the topic of automorphisms of noises and Arveson systems, and their link with the Arratia flow and sticky Brownian motion. B. Tsirelson gave a longer talk that contained an overview of the field and exposed several open problems. This was followed with a talk by J. Warren on perturbations of Arratia flows.

In the afternoon, D. Aldous presented several problems related to the study of geometrical paths defined from a Poisson field in $R_{d}$, for example the study of shortest cycles involving a fraction of the points in the Poisson cloud. Several open problems were presented. P. Bougerol reported on his work relating representation theory to path transformations. Y. Le Jan presented a new class of processes called relativistic diffusions, with special emphasis on the example of the Schwarzschild geometry.

Most of Tuesday was devoted to the study of random planar maps (that is, random planar graphs embedded in the plane). The day begun with a longer overview talk by O. Angel, who presented the model and some of the techniques introduced to study it (Matrix integrals, explicit bijections, recursions for generating functions). Angel explained some of the conjectured universal exponents and presented his work on triangulations of the unit disc and the relation of percolation on this object and the Airy stable process. The rest of the morning was devoted to three shorter talks by P. Chassaing, J.-F. Marckert and G. Miermont, who considered various aspects of the coding of planar maps by labelled trees, and the consequences of such coding for random planar quadrangulations and related models, and their scaling limits.

Tuesday afternoon began with a talk by J.-F. Le Gall who studied the conditioning of continuous trees in the framework of the so called “real trees”. He then applied his results to the study of objects appearing as scaling limits of encodings of planar maps. The rest of the afternoon was devoted to problems involving partial differential equations. J. Mattingly described his recent proof of ergodicity for stochastic Navier–Stokes under forcing by a finite number of modes. H. Owhadi discussed numerical schemes for the up-scaling of nonlinear divergence-form PDE's, based on metric deformations.

The subject of Wednesday's talks was the SLE and processes leading to SLE limits. G. Lawler discussed the Laplacian random walk, a generalization of the loop-erased random walk that should lead to an SLE limit, and sheds some light on how to define the SLE in non simply-connected domains. G. Kozma presented his proof of the existence of scaling limits for loop erased random walks in dimensions 2 and 3. J. Dubédat discussed aspects of the definition of multiple SLE's and their consequences for multiple crossing probabilities for some statistical physics models. R. Bauer described the possible generalizations of SLE in multiply connected domains via an analogue of Loewner's equations in such domains.

The weather on Wednesday afternoon was fine and the hike to St. Roman was pleasant.

The talks on Thursday focused on various models of motion in random media. In the morning, P. Mathieu presented his recent proof of diffusivity for random walk on the supercrtical percolation cluster, valid for any dimension greater than 2, and the two scale convergence ideas entering into the proof. M. Barlow, motivated by the study of random walk on the incipient infinite cluster of critical percolation in $Z_{d}$, studied this problem on trees, and described scaling limits for the distance traversed by the walk and related quantities. P. Mörters talked about the parabolic Anderson model and decsribed the four regimes that occur as a function of the tail of the driving noise. F. Castell described the limiting regimes for large deviations for random walks and Brownian motion in random sceneries.

In the afternoon, A.-S. Sznitman described the model of diffusions in random environments and presented a proof of diffusive behavior, in the perturbative regime and dimension at least $3$, for environments whose laws satisfy an isotropy condition. Y. Hu discussed limits of partition functions for oriented random polymer models. The rest of the afternoon was wrapped up by two talks on edge-reinforced random walks. In the first, F. Merkl introduced a description of linearly edge reinforced random walk on $Z_{2}$ in terms of a random walk in random environments, which might be a step in proving recurrence for the reinforced random walk. V. Limic discussed strongly reinforced walks on finite graphs, focusing on the case of a triangle and announcing the attractiveness of an edge under monotonicity conditions.

On Friday morning, T. Funaki discussed the dynamics of the Winterbottom droplet in a lattice model of coupled diffusions. This was followed by three shorter talks. The first, by A. Klenke, discussed the multi-fractal spectrum of intersection local time of two Brownian motions in dimensions 2 and 3. M. Winkel described natural growth processes with values in Lévy forests, that lead to natural limits. W. Koenig described a limit result for non-nearest neighbor random walks conditioned to remain ordered. He described Karlin–McGregor like formulae and convergence toward Dyson limits.

After lunch, T. Lyons concluded the meeting with a talk on rough paths and their signatures.

In addition to the talks during the day and the lively discussions throughout the day, there were two evening sessions in which current Ph.D. students presented briefly their results. These sessions included talks by J. Trujello-Ferreras, G. Pete, L. Georgen, T. Schmitz, C. Boutillier and A. Hammond.

## Cite this article

Wendelin Werner, Jean Dominique Deuschel, Ofer Zeitouni, Stochastic Analysis and Non-Classical Random Processes. Oberwolfach Rep. 2 (2005), no. 2, pp. 1231–1298

DOI 10.4171/OWR/2005/23