# Martin Barlow is the 2009 CRM-Fields-PIMS Prize Recipient

The Directors of the three Institutes, CRM, Fields and PIMS are pleased to announce that Martin Barlow from UBC is the recipient of the 2009 CRM-Fields-PIMS Prize. Martin Barlow is a leading figure in probability and the leading international expert in diffusion on fractals and other disordered media. In addition, the impact of his work has been important in such diverse fields as partial differential equations, including major progress on the De Giorgi conjecture, stochastic differential equations, the mathematical finance of electricity pricing, filtration enlargement and branching measure diffusions.

Already in the 1980’s, Martin Barlow settled a long open problem of probability theory, by providing necessary and sufficient conditions (the latter with J. Hawkes) for the continuity of local times of Lévy processes. This was the resolution of a thirty-year old problem which had attracted the efforts of Hale Trotter, Ronald Getoor and Harry Kersten among others. His conditions have paved the way for the study of the connection between local times and Gaussian processes.

In the 1990’s his detailed study of diffusions on a variety of fractals and fractal-like sets opened a new area of study in probability, making him the leading international expert in the behaviour of diffusions on fractals and other disordered media. The study of the diffusion on the Sierpinski carpet, started with Ed Perkins and then Richard Bass in 1986, served as a testing ground for diffusion in highly inhomogeneous media, a domain of interest for the physics community which is now within mathematical reach. Barlow remains at the leading edge of this research with his recent work giving best possible results for the behaviour of transition probabilities for random walks on super-critical percolation clusters. The pioneering papers on the diffusion on the Sierpinski carpet attracted to the domain experts in Dirichlet forms, diffusio