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

摘要

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 an 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.