Joint probability distributions for a class of non-Markovian processes

Area of Science:

• Physics
• Statistical Mechanics
• Stochastic Processes

Background:

• Coupled Langevin equations are fundamental in modeling complex systems.
• Existing research primarily focuses on single-time probability distributions.
• Understanding multi-time dynamics is crucial for non-Markovian systems.

Purpose of the Study:

• To derive and analyze N-time joint probability distributions for coupled Langevin equations.
• To extend existing probability distribution theories to higher-order temporal correlations.
• To investigate the connection between non-Markovian dynamics and fractional calculus.

Main Methods:

• Generalization of single-time probability distribution techniques to N-time distributions.
• Application of integral transform methods to relate non-Markovian to Markovian processes.
• Analysis of the integral kernel using partial differential equations with fractional time derivatives.

Main Results:

• A method for obtaining N-time joint probability distributions for coupled Langevin equations is presented.
• These distributions are shown to be derivable from Markovian process distributions via integral transforms.
• The integral kernel satisfies a fractional partial differential equation, indicating non-Markovian dynamics.

Conclusions:

• The study provides a novel framework for analyzing multi-time correlations in coupled Langevin systems.
• Fractional calculus naturally emerges when describing the non-Markovian nature of these systems.
• The findings offer new tools for studying complex systems exhibiting memory effects.