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Fractional relaxation noises, motions and the fractionalenergy balance equation
  • Shaun Lovejoy
Shaun Lovejoy
McGill University

Corresponding Author:lovejoy@physics.mcgill.ca

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We consider the statistical properties of solutions of the stochastic fractional relaxation equation that has been proposed as a model for the earth’s energy balance. In thisequation, the (scaling) fractional derivative term modelsenergy storage processes that occur over a wide range of space and time scales. Up until now, stochastic fractionalrelaxation processes have only been considered withRiemann-Liouville fractional derivatives in the context of random walk processes where it yields highlynonstationary behaviour. For our purposes we require the stationary processes that are the solutions of the Weyl fractional relaxation equations whose domain is −∞ to t rather than 0 to t. We develop a framework for handling fractional equationsdriven by white noise forcings. To avoid divergences, wefollow the approach used in fractional Brownian motion(fBm). The resulting fractional relaxation motions (fRm) and fractional relaxation noises (fRn) generalize the more familiar fBm and fGn (fractional Gaussian noise). Weanalytically determine both the small and large scale limitsand show extensive analytic and numerical results on the autocorrelation functions, Haar fluctuations and spectra. We display sample realizations. Finally, we discuss the prediction of fRn, fRm which – due to long memories - is a past value problem, not an initial value problem. We develop an analytic formula for the fRnforecast skill and compare it to fGn. Although the large scale limit is an (unpredictable) white noise that is attainedin a slow power law manner, when the temporal resolutionof the series is small compared to the relaxation time, fRncan mimick a long memory process with a wide range of exponents ranging from fGn to fBm and beyond. Wediscuss the implications for monthly, seasonal, annualforecasts of the earth’s temperature.