Stochastic Modelling for Levy Distributed Systems

The purpose of this paper is to examine a range of results that can be derived from Einstein’s evolution equation focusing on (but not in an exclusive sense) the effect of introducing a L´evy distribution. In this context, we examine the derivation (as derived from the Einstein’s evolution equation) of the classical and fractional diffusion equations, the classical and generalised Kolmogorov-Feller equations, the evolution of self-affine stochastic fields through the fractional diffusion equation and the fractional Schr¨odinger equation, the fractional Poisson equation (for the time independent case), and, a derivation of the Lyapunov exponent. In this way, we provide a collection of results (e.g. the derivation of certain partial differential equations) that are fundamental to the stochastic modelling associated with elastic scattering problems obtained under a unifying theme, namely, Einstein’s evolution equation. The approach is based on a multi-dimensional analysis of stochastic fields governed by a symmetric (zero-mean) Gaussian distribution and a L´evy distribution characterised by the L´evy index ∈ [0, 2].