Accurate path integral representations of the Fokker-Planck equation with a linear reference system: Comparative study of current theories

1998 ◽  
Vol 57 (1) ◽  
pp. 146-158 ◽  
Author(s):  
A. N. Drozdov ◽  
J. J. Brey
Keyword(s):  
Comparative Study ◽  
Path Integral ◽  
Reference System ◽  
Planck Equation ◽  
Integral Representations ◽  
Fokker Planck Equation ◽  
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.

scholarly journals Extended Mellin integral representations for the absolute value of the gamma function

Analysis ◽  
2018 ◽  
Vol 38 (1) ◽  
pp. 11-20
Author(s):  
Nicolas Privault
Keyword(s):  
Fourier Transform ◽  
Gamma Function ◽  
Bessel Functions ◽  
Planck Equation ◽  
Integral Representations ◽  
Modified Bessel Functions ◽  
Absolute Value ◽  
Fokker Planck Equation ◽  
The Absolute ◽  
Fokker Planck

AbstractWe derive Mellin integral representations in terms of Macdonald functions for the squared absolute value{s\mapsto|\Gamma(a+is)|^{2}}of the gamma function and its Fourier transform when{a<0}is non-integer, generalizing known results in the case{a>0}. This representation is based on a renormalization argument using modified Bessel functions of the second kind, and it applies to the representation of the solutions of a Fokker–Planck equation.

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