Master equations and the theory of stochastic path integrals

Area of Science:

• Stochastic Processes
• Mathematical Physics
• Computational Chemistry

Background:

• Master equations are crucial for understanding fluctuations in biological, chemical, and physical systems.
• Traditional analyses often rely on low-noise approximations, limiting accuracy for complex scenarios.
• A unified framework is needed to address master equations beyond these approximations.

Purpose of the Study:

• To provide a pedagogic introduction to master equations and path integral representations.
• To develop numerical and analytical methods that go beyond low-noise limits.
• To offer a unified framework for studying master equations and their applications.

Main Methods:

• Derivation of forward and backward master equations from the Chapman-Kolmogorov equation.
• Transformation of master equations into linear partial differential equations (PDEs).
• Expression of PDE solutions using forward and backward path integrals.

Main Results:

• Three detailed PDEs are presented, including novel ones for marginalized distributions and generating functionals.
• Two distinct path integral representations for conditional probability distributions are derived.
• Methods for computing rare event probabilities and analyzing continuous state space processes are discussed and extended.

Conclusions:

• The study offers a unified framework for analyzing master equations using path integrals, applicable to various complex systems.
• Novel methods provide accurate calculations beyond low-noise approximations, enhancing the study of fluctuations.
• The review bridges probability theory and concepts from quantum field theory, making advanced methods accessible.