Multidimensional Fokker-Planck Equation Research Articles

Generalized Onsager-Machlup function as solution of a multi-dimensional Fokker-Planck equation

The multi-dimensional Fokker-Planck equation is solved by Feynman path integrals. The solution may be interpreted as generalized Onsager-Machlup function.

Polarization Tracking of a Partially Coherent Signal Using a Double Loop

We consider the problem of simultaneously estimating the phase and polarization angles of a linearly polarized electromagnetic wave. The solution of this problem has applications to space communications and ionospheric probing. The nonlinear stochastic characteristics of a cross-coupled double-loop implementation for the tracker are evaluated using the multidimensional Fokker-Planck equation. The exact stationary probability density is found for a special case of equal signal-to-noise ratio (SNR) in the two loops. The stationary probability density is obtained for a general class of double loops when the two loop bandwidths differ greatly.

A sequence solution to the Fokker-Planck equation

In this paper a new approach to nonlinear system analysis based on the Fokker-Planck equation is developed. The development begins as a modification of the parametrix method of partial differential equation theory and provides a sequence solution to the multidimensional time-varying Fokker-Planck equation. This sequence is simplified for the case where only a steady-state solution or a solution for the variance of a particular combination of states is desired. Equations are presented in a form suitable for obtaining numerical results using a digital computer. This method is cast in a rather general mathematical framework and can provide useful results for a variety of problems. It is applied to the special case of phase-locked frequency demodulation, and curves of frequency tracking error are presented. Even though the rate of convergence of the sequence is not established in general, these numerical results prove the usefulness of the method for this special case and suggest a far greater range of usefulness.