Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman-Kac Approach

Area of Science:

• Mathematical Physics
• Stochastic Processes
• Reaction-Diffusion Systems

Background:

• Subdiffusive processes are common in various scientific fields but are challenging to model, especially when reactions are involved.
• Existing models often struggle with the long waiting times characteristic of subdiffusion.
• The Feynman-Kac equation provides a link between stochastic processes and partial differential equations.

Purpose of the Study:

• To derive a generalized Feynman-Kac equation applicable to subdiffusive processes with reactions.
• To introduce and define the concept of 'subordinated local time' for analyzing these processes.
• To establish a probabilistic framework for subdiffusion with reactions, connecting it to known boundary conditions.

Main Methods:

• Application of subordination techniques to derive a time fractional Schrödinger equation.
• Stochastic treatment to link the generalized Feynman-Kac equation with subdiffusive reaction-diffusion.
• Analytical solutions for first-reaction probability density, survival probability, and subordinated local time density.
• Validation through stochastic simulations.

Main Results:

• A generalized Feynman-Kac equation describing subdiffusion with reactions was derived.
• The subordinated local time was defined as the number of visits to a spatial point, independent of time spent.
• Equivalence was shown between the generalized Feynman-Kac equation with a reflecting boundary and the fractional diffusion equation with a radiation boundary.
• Analytic solutions in the time domain, expressed using the Wright function, were obtained for key probabilistic quantities.

Conclusions:

• The subordinated local time offers a novel and practical interpretation for subdiffusive processes with reactions.
• The derived generalized Feynman-Kac equation provides a powerful tool for analyzing complex reaction-diffusion phenomena.
• The study successfully bridges analytical mathematics and stochastic simulations, validating the new theoretical framework.