Fractional Stochastic Models for Heavy Tailed, and Long-Range Dependent, Fluctuations in Physical Systems

Abstract

Abstract A series of pioneering papers by Mandelbrot and his colleagues in the 1960s and early 1970s identified three effects present in many fluctuating time series drawn from observations in both the natural and the economic sciences, and put forward a mathematical model to describe each one. The effects were heavy tails in the amplitude of fluctuations, long range dependence, and autocorrelation between the absolute values of a time series. The effects have broad importance, with applications ranging across climate science in areas as diverse as the heavy tailed distributions of intense rainfall, the recurrence times between droughts, the clustering of extreme wind storms and the intermittent, bursty, volatility seen in fluid turbulence. Each effect exemplifies a very strong departure from one of the key properties of Gaussian white noise. They also greatly increased the visibility of the notions of self-similarity, power laws, and fractional calculus in physics. In this chapter I will summarise the three effects together, in order to compare and contrast them, and then go on to explain the reasons they were introduced, and the tools needed to understand and apply them. Introduction Why stochastic models for climate: At least two streams of theoretical investigation have contributed greatly to our present-day understanding of climate. One very visible one has been the global circulation models which solve fluid equations computationally, with appropriate closure schemes ( Houghton , 1986). Another stream, perhaps less visible outside the climate community but nonetheless very important, has been the more minimalist approaches, such as the energy balance models which encode as few assumptions as possible in order to study properties such as the latitudinal dependence of temperature ( Ghil , 1984). Both streams are deterministic, but the pioneering work of Hasselmann (1976) and Leith (1975) has led to a third group of approaches, reviewed by Gottwald et al. (2016), which face the irreducible complexity of the climate system by mixing deterministic and random components. These can be seen as modelling both the “centre of mass” and the fluctuations of a system's trajectory in the appropriate variable, such as temperature.