Front Propagation and Particle Systems (14w5055)


(Université UPMC - Paris VI)

Jeremy Quastel (University of Toronto)

Lenya Ryzhik (Stanford University)


The Banff International Research Station will host the "Front Propagation and Particle Systems" workshop from August 31st to September 5th, 2014.

Introduced in 1937 simultaneously by Fisher and Kolmogorov, Petrovskii and Piskunov, the celebrated F-KPP equation describes
various reaction-diffusion phenomena which can give rise to front propagation. It is a central feature of various models related to combustion, chemistry, biology and ecology (indeed, Fisher's original goal was to study how an advantageous allele spreads in a geographically distributed population).

Remarkably, this partial differential equation which is one of the simplest examples of a PDE that admits traveling wave solutions and which to a large extent started the field of semi-linear parabolic PDEs is also intimately connected with a class of probabilistic models, the {it branching random walks}, in a way
similar to the relation between the heat equation and the standard Brownian motion.
This connection, first studied by McKean in 1975, is at the heart of the famous result of Bramson (1983) which shows that
(1) for any initial condition decaying sufficiently fast, there is uniform convergence to the critical traveling wave solution, and that (2) the position of the front is growing linearly with a known logarithmic correction in time. The interplay between the F-KPP equation and branching
random walks was further exploited in several directions, both to study branching walks and the F-KPP equation itself.

The mathematical study of the problems surrounding the F-KPP equation has recently experienced a fast growth spurred by the progress both in the fields of PDEs and probability as well as in relation to modeling issues in various contexts (population dynamics, physiology, wound healing, tumor growth, etc.). Let us now outline some recent developments in the subject as well as some perspectives and open questions.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).