From theory of infinitely divisible distributions to derivation of generalized master equation for Markov process

Alexander A. Dubkov; Bernardo Spagnolo

doi:10.1117/12.500229

7 May 2003 From theory of infinitely divisible distributions to derivation of generalized master equation for Markov process

Alexander A. Dubkov, Bernardo Spagnolo

Author Affiliations +

Proceedings Volume 5114, Noise in Complex Systems and Stochastic Dynamics; (2003) https://doi.org/10.1117/12.500229
Event: SPIE's First International Symposium on Fluctuations and Noise, 2003, Santa Fe, New Mexico, United States

Abstract

We show that the increment of generalized Wiener process (random process with stationary and independent increments) has the properties of a random value with infinitely divisible distribution. This enables us to write the characteristic function of increments and then to obtain the new formula for correlation of the derivative of generalized Wiener process (non-Gaussian white noise) and its arbitrary functional. IN the context of well-known functional approach to analysis of nonlinear dynamical systems based on a correlation formulae for nonlinear stochastic functionals, we apply this result for derivation of generalized Fokker-Planck equation for probability density. We demonstrate that the equation obtained takes the form of ordinary Fokker-Planck equation for Gaussian white noise and, at the same time, transforms in the fractional diffusion equation in the case of non-Gaussian white noise with stable distribution.

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Alexander A. Dubkov and Bernardo Spagnolo "From theory of infinitely divisible distributions to derivation of generalized master equation for Markov process", Proc. SPIE 5114, Noise in Complex Systems and Stochastic Dynamics, (7 May 2003); https://doi.org/10.1117/12.500229

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