Abstract
The steady-state correlation functions of non-linear stochastic processes driven by parametric noise are studied. A systematic method proposed by Nadler and Schulten is applied beyond the lowest order for the first time in this context. It is reformulated in a way which admits generalization to noises other than Gaussian and white. The explicit results obtained close to the instability point for the Verhulst model with Gaussian white noise improve considerably those previously obtained by continued-fraction techniques.
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The relations (2.4) and (2.6) can be easily obtained by expandingC(t) or exp(−ωt) of (2.2) respectively in powers oft
The truncated continued fraction expansion is nothing but a rational function of ω which coincides with the exact Laplace transformC(ω) in a given number of the first coefficients of the 1/ω-expansion
In (2.11) it is implicitly assumed that the processx(t) is Markovian. For non-Markovian processes the method is still valid when reformulated in its equivalent multivariable Markovian form. For more details see[30]
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It has been argued that for additive noise models, a divergence of a relaxation time can only occur in the deterministic limit (D→0) as the so-called asymptotic critical slowing down of [30]
In general one could characterize long-time tails decaying with different exponents according to which relaxation moment diverges first
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Casademunt, J., Hernández-Machado, A. Correlation functions near instabilities in systems driven by parametric noise. Z. Physik B - Condensed Matter 76, 403–411 (1989). https://doi.org/10.1007/BF01321919
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DOI: https://doi.org/10.1007/BF01321919