Derivation of Feynman–Kac and Bloch–Torrey Equations in a Trapping Medium

Abstract

We derive rigorously the fractional counterpart of the Feynman–Kac equation for a transport problem with trapping events characterized by fat-tailed time distributions. Our starting point is a random walk model with an inherent time subordination process due to the difference between clock times and operational times. We pass to the hydrodynamic limit using weak convergence arguments. Due to the lack of regularity of physical data, we use the framework of measure theory. We finally derive a Bloch–Torrey type equation for nuclear magnetic resonance data in this subdiffusive context.