Restricted diffusion and related inverse problems

by D. S. Grebenkov

Laboratoire de Physique de la Matiere Condensee,
CNRS -- Ecole Polytechnique, F-9128 Palaiseau France


Abstract

We present recent advances in studying the dynamics of 
nuclei diffusing in a bounded domain with a (partially) 
reflecting boundary. Application of inhomogeneous magnetic 
fields allows one to "encode" stochastic trajectories of 
the nuclei and therefore to monitor the dynamics experimentally. 
This so-called diffusion-weighted imaging has found numerous 
applications in medicine and oil-recovery industry. In 
probabilistic terms, this is equivalent to studying an 
integral functional of reflected Brownian motion. In PDE 
context, the problem is described by diffusion (or heat) 
equation with a complex-valued spatially inhomogeneous 
reaction rate. Using the Laplace operator eigenbasis, we 
retrieve perturbative results in the short-time and long-time 
limits. We also consider the related inverse problems and 
discuss what information about an unknown geometry of the 
confinement can be extracted from such a diffusion-weighted 
imaging? Given the spectral formulation, this inverse problem 
revives the Kac's question "Can one hear the shape of a drum?" 
in a new context. 

Reference:

D. G. Grebenkov, NMR Survey of Reflected Brownian Motion,
Reviews of Modern Physics 79, 1077 (2007).