Fundamental solution of the space-time fractional diffusion equation using Monte Carlo simulations of CTRWs


Friday, 12 July, 2024 - 15:00 to 16:00

Speaker : Enrico Scalas, Department of Statistical Sciences / Sapienza University of Rome, Italy

Time : 15:00 - 16.00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

Zoom : A link will appear here, an hour before the talk.

Organizers : Pavan Pranjivan Mehta* ( and Arran Fernandez** (

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: Space-time fractional diffusion equation, Continuous-time random walks, Monte Carlo simulations

Abstract: I present a numerical method for the Monte Carlo simulation of uncoupled continuous-time random walks with a L ́evy α-stable distribution of jumps in space and a Mittag–Leffler distribution of waiting times, and apply it to the stochastic solution of the Cauchy problem for a partial differential equation with fractional derivatives both in space and in time. The one-parameter Mittag–Leffler function is the natural survival probability leading to time-fractional diffusion equations. Transformation methods for Mittag Leffler random variables were found later than the well-known transformation method by Chambers, Mallows, and Stuck for L ́evy α–stable random variables and they are now widely used. Combining the two methods, one obtains an accurate approximation of space– and time–fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes. If time allows, I will discuss directions for further research. This was joint work with Daniel Fulger and Guido Germano [1].

I delivered a five-hour course on this topic at the on–line UG LMS Summer School held in Swansea in 2021 [2].

Biography: Enrico Scalas is professor of probability and mathematical statistics at the Department of Statistical Sciences of Sapienza, University of Rome. He is co-author of more than 140 publications including two monographs and an edited book. If you like bibliometry, you’ll be delighted to know that his works received more than 11000 citations and his h index is 43 (according to Google Scholar). Otherwise, you may just be interested in the fact that his research concerns the foundations of economics, finance and statistical physics and, for this reason, he became a probabilist working on various stochastic processes often with fractional flavour.

For the time being, a rather complete list of his publications up to 2022 can be found at:

More recent publications are listed at, following the link Profilo Research - Pubblicazioni IRIS


[1] Daniel Fulger, Guido Germano, and Enrico Scalas. “Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation”. In: Physical Review E 77 (2008), p. 021122.

[2] YouTube. “Enrico Scalas: Monte Carlo simulations of anomalous random walks”. In: (2021).