Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



A suggested modification of noise theory

Author: Julian Keilson
Journal: Quart. Appl. Math. 12 (1954), 71-76
MSC: Primary 60.0X
DOI: https://doi.org/10.1090/qam/82754
MathSciNet review: 82754
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Abstract: A class of stationary, equilibrium, Markoff processes is demonstrated all of which have the same equilibrium distribution, $ {W_0}\left( x \right)$, and correlation function $ R\left( t \right) = {E_0}\exp \left( { - t/{\tau _0}} \right)$ differing from each other in the number of zero crossings of the system per second. The processes are described by an integral equation characterized by a parameter $ \gamma $. As $ \gamma $ approaches 1, the integral equation passes over into the Fokker-Planck equation $ {\tau _0}\frac{{\partial W}}{{\partial t}} = {E_0}\frac{{{\partial ^2}W}}{{\partial {x^2}}} + \frac{\partial }{{\partial x}}\left( {xW} \right)$. Since the number of zero crossings per second of the system becomes infinite as $ \gamma $ goes to one, the degenerate nature of the Fokker-Planck process is made evident.

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DOI: https://doi.org/10.1090/qam/82754
Article copyright: © Copyright 1954 American Mathematical Society

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