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Generalizations and Extensions of the Fokker-Planck-Kolmogorov Equations


Pawula, Robert Francis (1965) Generalizations and Extensions of the Fokker-Planck-Kolmogorov Equations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/K713-DX65.


The problem of determining probability density functions of general transformations of random processes is considered in this thesis. A method of solution is developed in which partial differential equations satisfied by the unknown density function are derived. These partial differential equations are interpreted as generalized forms of the classical Fokker-Planck-Kolmogorov equations and are shown to imply the classical equations for certain classes of Markov processes. Extensions of the generalized equations which overcome degeneracy occurring in the steady-state case are also obtained.

The equations of Darling and Siegert are derived as special cases of the generalized equations thereby providing unity to two previously existing theories. A technique for treating non-Markov processes by studying closely related Markov processes is proposed and is seen to yield the Darling and Siegert equations directly from the classical Fokker-Planck-Kolmogorov equations.

As illustrations of their applicability, the generalized Fokker-Planck-Kolmogorov equations are presented for certain joint probability density functions associated with the linear filter. These equations are solved for the density of the output of an arbitrary linear filter excited by Markov Gaussian noise and for the density of the output of an RC filter excited by the Poisson square wave. This latter density is also found by using the extensions of the generalized equations mentioned above. Finally, some new approaches for finding the output probability density function of an RC filter-limiter-RC filter system driven by white Gaussian noise are included. The results in this case exhibit the data required for complete solution and clearly illustrate some of the mathematical difficulties inherent to the use of the generalized equations.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:(Electrical Engineering)
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Electrical Engineering
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Braverman, David J.
Thesis Committee:
  • Unknown, Unknown
Defense Date:16 February 1965
Funding AgencyGrant Number
Hughes Aircraft CompanyUNSPECIFIED
Naval Ordnance Test StationN123(60530)34804A
Naval Ordnance Test StationN123(60530)51581A
Record Number:CaltechTHESIS:03182015-145729313
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:8789
Deposited By: Benjamin Perez
Deposited On:19 Mar 2015 15:30
Last Modified:21 Feb 2024 18:45

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