Tempered fractional Feynman-Kac equation: Theory and examples

Area of Science:

• Physics
• Mathematics
• Statistical Mechanics

Background:

• Functionals of Brownian and non-Brownian motions are crucial in various scientific fields.
• Anomalous diffusion, particularly continuous time random walks, exhibits complex behaviors.
• Understanding the statistical properties of diffusion processes is essential.

Purpose of the Study:

• To derive forward and backward fractional Feynman-Kac equations.
• To describe the distribution of functionals for space and time-tempered anomalous diffusion.
• To analyze specific functionals including occupation time, first passage time, and maximal displacement.

Main Methods:

• Derivation of fractional Feynman-Kac equations.
• Analysis of continuous time random walk models.
• Explicit treatment of various functionals.

Main Results:

• Established fractional Feynman-Kac equations for anomalous diffusion.
• Provided descriptions for distributions of functionals.
• Investigated functionals such as occupation time in half-space and first passage time.

Conclusions:

• The derived equations offer a powerful tool for studying anomalous diffusion.
• The findings enhance the understanding of functionals in complex diffusion processes.
• This work contributes to the theoretical framework of continuous time random walks.