Journal
PROBABILITY THEORY AND RELATED FIELDS
Volume 127, Issue 3, Pages 305-352Publisher
SPRINGER-VERLAG
DOI: 10.1007/s00440-003-0284-0
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A fractal field is a collection of fractals with, in general, different Hausdorff dimensions, embedded in R-2. We will construct diffusion processes on such fields which behave as Brownian motion in R-2 outside the fractals and as the appropriate fractal diffusion within each fractal component of the field. We will discuss the properties of the diffusion process in the case where the fractal components tile R-2. By working in a suitable shortest path metric we will establish heat kernel bounds and large deviation estimates which determine the trajectories followed by the diffusion over short times.
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