A path integral method for coarse-graining noise in stochastic differential equations with multiple time scales |
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Authors: | Tobias Schäfer Richard O Moore |
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Institution: | a Department of Mathematics, The College of Staten Island, City University of New York, NY, United Statesb Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, United States |
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Abstract: | We present a new path integral method to analyze stochastically perturbed ordinary differential equations with multiple time scales. The objective of this method is to derive from the original system a new stochastic differential equation describing the system’s evolution on slow time scales. For this purpose, we start from the corresponding path integral representation of the stochastic system and apply a multi-scale expansion to the associated path integral kernel of the corresponding Lagrangian. As a concrete example, we apply this expansion to a system that arises in the study of random dispersion fluctuations in dispersion-managed fiber-optic communications. Moreover, we show that, for this particular example, the new path integration method yields the same result at leading order as an asymptotic expansion of the associated Fokker-Planck equation. |
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Keywords: | Multi-scale analysis Coarse-graining of noise Fiber optics |
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