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New Scaling Limits workshop: abstracts

Abstracts for the four minicourses

• Jean Bertoin: Coalescence and stochastic flows of bridges

I will present a series of joint works [1,2,3] with Jean-Francois Le Gall in which the connection between exchangeable coalescents (also known as coalescents with multiple collisions) and certain stochastic flows of bridges is developed. Exchangeable coalescents have been introduced as models for the genealogy of large populations with constant size [5,7,8], and include the celebrated coalescent of Kingman [4] as a most important example. The connection with stochastic flows yields several interesting applications, for instance to the asymptotic behavior of these coalescents, and to an interpretation of a version of Smoluchowski's coagulation equation [6] with multiple coagulations.

[1] Bertoin, J. and Le Gall, J.-F. "Stochastic flows associated to coalescent processes." Probab. Theory Relat. Fields 126 (2003), 261-288.
[2] Bertoin, J. and Le Gall, J.-F. "Stochastic flows associated to coalescent processes, II: Stochastic differential equation." Ann. Inst. Henri Poincaré, Probabilités et Statistiques 41 (2005), 307-333.
[3] Bertoin, J. and Le Gall, J.-F. "Stochastic flows associated to coalescent processes, III: Limits theorems." Illinois J. Math. 50 (2006), 147-181
[4] Kingman, J. F. C. "The coalescent." Stochastic Process. Appl. 13 (1982), 235-248.
[5] Moehle, M. and Sagitov, S. "A classification of coalescent processes for haploid exchangeable population models." Ann. Probab. 29 (2001), 1547-1562.
[6] Norris, J. R. "Smoluchowski's coagulation equation: uniqueness, non-uniqueness and hydrodynamic limit for the stochastic coalescent." Ann. Appl. Probab. 9 (1999), 78109.
[7] Pitman, J. "Coalescents with multiple collisions." Ann. Probab. 27 (1999), 1870-1902.
[8] Schweinsberg, J. "Coalescents with simultaneous multiple collisions." Electron. J. Probab. 5-12 (2000), 1-50.

• Franco Flandoli: SPDEs in fluid dynamics

1: Example of a non well posed linear PDE that becomes well-posed under random perturbations

It is well known that certain non well posed ODEs become well posed when non degenerate noise is added. A new proof of this fact is given providing existence of a differentiable stochastic flow for a class of non-Lipschitz drift. Based on this result, one can solve a linear transport equation with multiplicative noise (an SPDE) in spite of the fact that the same equation without noise in non well posed. It is interesting to notice that improvement of well-posed-ness is caused by non degenerate additive type noise for ODEs, while multiplicative noise looks better for SPDEs.

2: Partial progresses on well posedness of 3D stochastic Navier-Stokes equations

In 2003, Da Prato and Debussche gave a direct solution of the Kolmogorov equation associated to 3D stochastic Navier-Stokes equations. Among the interesting byproducts, existence of a strong Feller Markov semigroup must be mentioned. But uniqueness remains an open problem. The lecture reviews this subject under the viewpoint developed later by Flandoli and Romito, including a result of equivalence of transition probabilities among possibly different Markov selections.

3: Scaling exponents in SPDEs

A formal computation based on a scaling argument seems to indicate that zero viscosity stochastic Navier-Stokes equations forced by infinitely large scale noise possesses the scaling exponent 2/3 for the so called second order structure function. One can show that such a result is not true in two dimensions. In dimension three it looks closer to reality but a correction is presumably needed. A review of the few rigorous results and the many open problems on this subject is presented, comparing also with other nonlinear SPDEs like a stochastic GOY model or Prouse equation with velocity-dependent viscosity.

• Kurt Johansson: Scaling limits in random matrix theory and related models

(abstract not yet available)

• Scott Sheffield: Random geometry and the Schramm-Loewner evolution

Many "quantum gravity" models in mathematical physics can be interpreted as probability measures on the space of metrics on a Riemannian manifold. We describe several recently derived connections between these random metrics and certain random fractal curves called "Schramm-Loewner evolutions" (SLE).