Fractional Feynman-Kac equation for non-brownian functionals.

Area of Science:

• Physics
• Mathematics
• Statistical Mechanics

Background:

• Anomalous diffusion describes particle movement deviating from standard Brownian motion.
• Fractional calculus offers a framework to model non-local and long-range dependent processes.
• Feynman-Kac equations link stochastic processes to partial differential equations.

Purpose of the Study:

• To derive backward and forward fractional Feynman-Kac equations for anomalous diffusion.
• To analyze the distribution of functionals of particle paths in anomalous diffusion scenarios.
• To investigate the relationship between anomalous diffusion statistics and ergodicity breaking.

Main Methods:

• Utilizing fractional substantial derivatives as introduced by Friedrich et al.
• Deriving fractional backward and forward Feynman-Kac equations.
• Calculating the distribution of occupation times in a half-space domain.

Main Results:

• Successfully derived fractional Feynman-Kac equations for anomalous diffusion.
• Obtained the distribution of occupation times in a half space.
• Established a connection between anomalous functional statistics and weak ergodicity breaking.

Conclusions:

• Fractional substantial derivatives provide a suitable mathematical framework for anomalous diffusion.
• The derived equations offer a novel approach to studying path-dependent functionals.
• The findings shed light on the nature of weak ergodicity breaking in anomalous processes.