Self-similarity and response of fractional differential equations under white noise input

Self-similarity, fractal behaviour and long-range dependence are observed in various branches of physical, biological, geological, socioeconomics and mechanical systems. Self-similarity, also termed self-affinity, is a concept that links the properties of a phenomenon at a certain scale with the same properties at different time scales as it happens in fractal geometry. The fractional Brownian motion (fBm), i.e. the Riemann-Liouville fractional integral of the Gaussian white noise, is self-similar; in fact by changing the temporal scale t -> at (a > 0), the statistics in the new time axis (at) remain proportional to those calculated in the previous axis (t). The proportionality coefficient is a(2H) being H > 0 the Hurst index. In the practical applications, the phenomena are usually ruled by fractional differential equations involving more terms. In this paper it is shown that the response of a multi-term fractional differential equation is a linear combination of self-similar processes with increasing order of Hurst exponent. The consequences of self-similarity are discussed in detail, closed forms of correlation and variance are presented for the general case and particularized for the cases of engineering interest.