Generating functional approach to space- and time-dependent colored noise

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Abstract
A stochastic model for time- and space-dependent colored noise is suggested based on the following assumptions: (1) the noise field is generated by a random number of point events corresponding to a correlated point process; (2) the contributions of the different events to the noise field are additive, each contribution being a random function selected from the same probability density functional. An analytical treatment of the noise field described by these assumptions is possible; the generating functional of the noise field as well as the corresponding cumulants can be computed exactly. All cumulants are explicitly evaluated when the contribution of an event is given by a diffusion equation. A detailed analysis of the asymptotic behavior is made for time-homogeneous and translationally invariant processes. A Gaussian random field colored in space and time emerges for very frequent independent events of very small intensities, provided that the central moments of the contribution of an event to the noise field are finite and fast decreasing in space. The correlations among events lead to corrections of the Gauss limit law; however, the cumulants of the random fields are also finite. If the central moments of an event are slowly decreasing in space a different type of asymptotic behavior occurs; the cumulants of the noise field become infinite and the resulting field is described by a non-Gaussian stable law of the Lévy type. The theory may be applied to the study of stochastic gravitational fluctuations in galactic systems, to the analysis of concentration fluctuations for diffusion processes in disordered systems and for the analysis of the influence of environmental fluctuations in continuum mechanics and electrodynamics.

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