Generalized Fokker-Planck equation for a class of stochastic dynamical systems driven by additive Gaussian and Poissonian fractional white noises of order <Emphasis Type="Italic">α</Emphasis> |
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Authors: | Guy Jumarie |
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Institution: | (1) Department of Mathematics, University of Québec at Montréal, P.O. Box 8888, Downtown Station, Montréal, Qc H3C 3P8, Canada |
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Abstract: | In a first stage, the paper deals with the derivation and the solution of the equation of the probability density function
of a stochastic system driven simultaneously by a fractional Gaussian white noise and a fractional Poissonian white noise
both of the same order. The key is the Taylor’s series of fractional order f(x + h) = E
α(hαD
x
α)f(x) where E
α() denotes the Mittag-Leffler function, and D
x
α is the so-called modified Riemann-Liouville fractional derivative which removes the effects of the non-zero initial value
of the function under consideration. The corresponding fractional linear partial differential equation is solved by using
a suitable extension of the Lagrange’s technique involving an auxiliary set of fractional differential equations. As an example,
one considers a half-oscillator of fractional order driven by a fractional Poissonian noise.
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Keywords: | fractional Gaussian noise fractional Poissonian noise fractional partial differential equation fractional Fokker Planck equation fractional stochastic differential equations |
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