Random Walk Models for Space-fractional Diiusion Processes Random Walk Models for Space-fractional Diiusion Processes

By space-fractional (or L evy-Feller) diiusion processes we mean the processes governed by a generalized diiusion equation which generates all L evy stable probability distributions with index (0 < 2), including the two symmetric most popular laws, Cauchy (= 1) and Gauss (= 2). This generalized equation is obtained from the standard linear diiusion equation by replacing the second-order space derivative with a suitable fractional derivative operator, deened as inverse of the Feller potential (a generalization of the Riesz potential). In this paper, excluding the singular case = 1 and based on the Gr unwald-Letnikov approach to the fractional derivative, we propose to approximate these processes by random walk models, discrete in space and time. It is proved that for properly scaled transition to the limit of vanishing space and time steps there is convergence to all (symmetric and nonsymmetric) stable distributions (evolving in time). All stable distributions are thus obtained, with the exception of the Cauchy distribution which resists to this treatment.