Abstract
We derive an equation of the Chapman-Kolmogorov type for discrete multi-dimensional mappings under the action of additive and multiplicative noise with arbitrary distribution function. The resulting equation is reduced to a Fredholm integral equation. By iteration of the Chapman-Kolmogorov equation as usual, a path integral solution is found. Specializing the distribution function of the noise to a Gaussian distribution and taking the Fourier transform contant can be made with the path integral formulation used by Shraiman, Wayne and Martin.
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Haken, H., Mayer-Kress, G. Chapman-Kolmogorov equation and path integrals for discrete chaos in presence of noise. Z. Physik B - Condensed Matter 43, 185–187 (1981). https://doi.org/10.1007/BF01293609
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DOI: https://doi.org/10.1007/BF01293609