Abstract

We derive, through subordination techniques, a generalized Feynman–Kac equation in theform of a time fractional Schrödinger equation. We relate such equation to a functional whichwe name the subordinated local time. We demonstrate through a stochastic treatment how thisgeneralized Feynman–Kac equation describes subdiffusive processes with reactions. In thisinterpretation, the subordinated local time represents the number of times a specific spatialpoint is reached, with the amount of time spent there being immaterial. This distinctionprovides a practical advance due to the potential long waiting time nature of subdiffusiveprocesses. The subordinated local time is used to formulate a probabilistic understandingof subdiffusion with reactions, leading to the well known radiation boundary condition.We demonstrate the equivalence between the generalized Feynman–Kac equation with areflecting boundary and the fractional diffusion equation with a radiation boundary. Wesolve the former and find the first-reaction probability density in analytic form in the timedomain, in terms of the Wright function. We are also able to find the survival probabilityand subordinated local time density analytically. These results are validated by stochasticsimulations that use the subordinated local time description of subdiffusion in the presenceof reactions.