“The paper describes a new discovery, the limiting behavior of continuous  time random walks with infinite mean waiting times.”

The paper describes a limit theorem for continuous-time random walks with infinite mean waiting times. This is a simple but quite useful model with connections to the emerging theory of fractional partial differential equations.

The limiting behavior of continuous-time random walks with infinite mean waiting times is a new discovery. The paper also surveys some interesting connections with fractional calculus.

The basic model—continuous-time random walk—supposes that a particle waits a random amount of time between jumps, and that the subsequent jump sizes are also random. One application is for particles of a pollutant being carried along in a stream or underground aquifer.

Another application is in finance, where the particle is the price of a stock, the jumps are changes in price, and the waiting times account for the time between trades.

We got interested in this work after we were approached by a scientist, Dr. David A. Benson of the Department of Geology and Geological Engineering at the Colorado School of Mines, with some questions about fractional partial differential equations.

We eventually discovered that certain random walk models were a stochastic foundation for these problems. The connection between random models and their deterministic counterparts led to a number of interesting discoveries. The main problem for us, as probabilists, was to try and understand the problems in fractional calculus and the applications in hydrology. The greatest pleasure, on the other hand, was to see mathematical theory applied to real problems in science.

This particular model is important for pollution control, especially for organic pollutants that stick to the soil. The waiting time until a particle releases from the soil is longer than previously thought, which creates problems for the cleanup of toxic spills.

We hope that the new fractional partial differential equation models for these sticking particles will lead to more realistic predictions of cleanup time.