brownian motion on fractals

Brownian motion on fractals is a recent addition to the fauna of mathematical studies.

Writes Martin Barlow:
The initial motivation for this area came from physicists working in the theory of condensed matter, who consider fractals to be good models of some objects (such as polymers) which arise there, and who were interested in `transport properties' of these objects, such as conduction of heat or transmission of vibrations. It was therefore natural to wish to study equations such as the heat equation on fractal spaces, and to do this one has to define and study a `Laplacian operators' on fractal spaces.

Mathematicians have become interested in the field more recently, and have studied both the natural diffusion processes on these spaces, and the analysis associated with the natural Laplacian. In [*] a very detailed study was made of Brownian motion on the simplest non-trivial fractal space, the Sierpinski gasket. Following this, with Richard Bass I have studied the richer and more complicated class of Sierpinski carpets.

Fractals provide a new area to explore the ideas associated with harmonic analysis, but one where many of the standard tools fail to work. Thus there is an incentive to develop new techniques - an example would be the use of Dirichlet forms.

[*] "Necessary and sufficient conditions for the continuity of local time of Levy processes." Annals of Probability, 16, 1389-1427 (1988).

My own research is on processes on Simple Fractal spaces, a class of fractals which are characterized by that each cell has two connection points. The study of Brownian motion on this class is the first which included fractals with non-uniform scaling - an excellent study incling fractals with non-uniform scaling has later been done by Fitzsimmons, Hambly and Kumagai ("Affine fractals"), and my study is also the first construction of Brownian motion on non-symmetric fractals.