Subdiffusive master equation with space dependent anomalous exponent: `Black Swan' effects

Abstract

We derive the fractional master equation with space dependent anomalous exponent. We analyze the asymptotic behavior of corresponding lattice model both analytically and by Monte Carlo simulation. We show that the subdiffusive fractional equations with constant anomalous exponent $\mu $ in a bounded domain $[ 0,L]$ are not structurally stable with respect to the non-homogeneous variations of parameter $\mu $. In particular, the Gibbs-Boltzmann distribution is no longer the stationary solution of the fractional Fokker-Planck equation whatever the space variation of the exponent might be. We analyze the random distribution of $% \mu$ in space and find that in the long time limit, the probability distribution is highly intermediate in space and the behavior is completely dominated by very unlikely events.