
I am a PhD student of the Mathematics department at Bar Ilan University. My PhD is in probability and titled "On the Properties of the Fractional Poisson Process and Fractional Poisson Fields". More Broadly, I investigate Continuous Time Random Walk Limits (CTRWL). CTRW is a random walk coupled with random waiting times such that each jump in space is followed (or preceded) by a random waiting time. A typical path of a CTRW looks like this
Limit procedure on the CTRW yields a CTRWL, which is generally an anomalous diffusion. More specifically, when the waiting times are assumed to be heavytailed the CTRWL exhibits a subdiffusive behavior. For example, when the jumps in space are centered with finite variance and the waiting times are heavy tailed then we get something that looks like this
You can see in the picture the intervals in which the diffusion is immobile. This is characteristic of CTRWL and indicative of particles that are stuck due to long waiting times. In the video below you can check out the convergence of the CTRW (first pic) to the CTRWL (second pic).