Collaborative Research: Stochastic Methods for Fractional Partial Differential Equations

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Brownian motion, the scaling limit of a simple random walk, is a stochastic model for diffusion. Probability densities for this stochastic process solve the classical diffusion equation. A continuous time random walk is a simple random walk subordinated to a renewal process, used in physics to model anomalous diffusion. Increments of the simple random walk represents particle jumps, and the renewal epochs represent the particle jump times. Infinite variance particle jumps cause superdiffusion, in which a cloud of particles spreads faster than the classical model predicts. Infinite mean waiting times lead to subdiffusion. Scaling limits of continuous time random walks are stochastic processes whose densities solve fractional partial differential equations. Infinite variance particle jumps lead to operator stable Levy motions, while infinite mean waiting times induce subordination to an inverse stable subordinator. Fractional derivatives are generators of stable continuous convolution semigroups. The research funded by this grant is developing a sound mathematical basis for this physical theory, and pursuing practical applications to problems in contaminant transport. The goals of this research include limit theory for continuous time random walks with possible dependence between the jump sizes and the waiting times, extension to more realistic multidimensional jump vectors with matrix scaling to allow different rates of particle spreading in each coordinate, parallel development of multiscaling fractional derivative operators in space, analysis of the fundamental physical basis of fractional diffusion to elucidate the physical meaning of the model parameters, development of useful statistical methods for parameter estimation, numerical methods for fractional partial differential equations, and application of these methods to real data from laboratory and field experiments and remediation efforts involving porous media and fracture flow. Movement of contaminants in a mountain stream takes place over a vast range of time scales. A majority will move away quickly, but a small amount can be caught in eddies for minutes or days. Another fraction may move into the relatively motionless water beneath the streambed, and it may take months or years for the last molecules to disappear completely. The movement of contaminants in underground water spreads over a similar range of time scales. Our team of mathematicians and hydrogeologists uses modern interdisciplinary research methods to develop accurate models for the movement and spread of these contaminants. Much more than an academic exercise, this work is needed for realistic models of contaminant and nutrient movement in drinking water supplies. The research is necessary because existing diffusion models greatly underestimate the time and concentration at which contaminants arrive downstream, when compared to actual data.