Fractional diffusion equations and processes with randomly varying time

Abstract

In this paper the solutions $u_{\nu}=u_{\nu}(x,t)$ to fractional diffusion equations of order $0<\nu \leq 2$ are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order $\nu =\frac{1}{2^n}$, $n\geq 1,$ we show that the solutions $u_{{1/2^n}}$ correspond to the distribution of the $n$-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order $\nu =\frac{2}{3^n}$, $n\geq 1,$ is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that $u_{\nu}$ coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions $u_{\nu}$ and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.Comment: Published in at http://dx.doi.org/10.1214/08-AOP401 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Diffusion in multiscale spacetimes

“…It does not depend on (i ) the measure weight w(k) in momentum space, nor on (ii ) the relative sign between diffusion and kinetic operators (hyperbolic or parabolic diffusion equation), nor on (iii ) the presence of friction or source terms, as one can convince oneself by a direct inspection (e.g., [86,104] and references therein). Consequently, all these properties will be inherited by the spectral dimension.…”

“…Another formulation, more convenient to study the analytic properties of P , is the following [86]. In one dimension, if 6 Often it is also described by another transport equation, called a bifractional or fractional Fick equation, where the diffusion equation is "redistributed," (∂ σ − ∂ 1−β σ ∂ 2 x )P = 0 [58,61].…”

“…An example of particular importance for us is iterated Brownian motion (IBM) [86][87][88][89][90][91][92][93][94][95][96][97][98][99][100][101][102][103][104]. To illustrate iterated Brownian motion, we set α = 1 and consider the ordinary D-dimensional higher-order operator ∇ n , Eq.…”

“…In Sec. V B we shall consider telegraph processes [85,86,98,105,106]. A remarkable duality shows that fractional and iterated Brownian motions can be identified.…”

“…The return probability is defined as the spatial average of P , i.e., 8 Parabolic equations of the type (87) were studied in Refs. [86,[104][105][106][110][111][112] and generalized to higher-order operators ∇ n x and several other forms. the trace of the heat kernel per unit volume,…”