Generalized Onsager-Machlup function and classes of path integral solutions of the Fokker-Planck equation and the master equation

1976 ◽  
Vol 24 (3) ◽  
pp. 321-326 ◽  
Author(s):  
H. Haken
Keyword(s):  
Master Equation ◽  
Path Integral ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Integral Solutions ◽  
Fokker Planck

Simulating Drift-Diffusion Processes With Generalized Interval Probability

2012 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Path Integral ◽  
Stochastic Systems ◽  
Diffusion Processes ◽  
Epistemic Uncertainty ◽  
Planck Equation ◽  
Integral Approach ◽  
Drift Diffusion ◽  
Interval Probability ◽  
Fokker Planck Equation ◽  
Fokker Planck

The Fokker-Planck equation is widely used to describe the time evolution of stochastic systems in drift-diffusion processes. Yet, it does not differentiate two types of uncertainties: aleatory uncertainty that is inherent randomness and epistemic uncertainty due to lack of perfect knowledge. In this paper, a generalized Fokker-Planck equation based on a new generalized interval probability theory is proposed to describe drift-diffusion processes under both uncertainties, where epistemic uncertainty is modeled by the generalized interval while the aleatory one is by the probability measure. A path integral approach is developed to numerically solve the generalized Fokker-Planck equation. The resulted interval-valued probability density functions rigorously bound the real-valued ones computed from the classical path integral method. The new approach is demonstrated by numerical examples.

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