Modern Stochastics: Theory and Applications

The first-return time is the time that it takes a random walker to go back to the initial position for the first time. In this paper, the first-return time is studied when random walkers perform fractional kinetics, specifically fractional diffusion, that is modelled within the framework of the continuous-time random walk on homogeneous space in the uncoupled formulation with Mittag-Leffler distributed waiting-times. Both the Markovian and non-Markovian settings are considered, as well as any kind of symmetric jump-size distributions, namely with finite or infinite variance. It is shown that the first-return time density is indeed independent of the jump-size distribution when it is symmetric, and therefore it is affected only by the waiting-time distribution that embodies the memory of the process. The analysis is performed in two cases: first jump then wait and first wait then jump, and several exact results are provided, including the relation between results in the Markovian and non-Markovian settings and the difference between the two cases.

A fractional advection-dispersion equation (fADE) has been advocated for heavy-tailed flows where the usual Brownian diffusion models fail. A stochastic differential equation (SDE) driven by a stable Lévy process gives a forward equation that matches the space-fractional advection-dispersion equation and thus gives the stochastic framework of particle tracking for heavy-tailed flows. For constant advection and dispersion coefficient functions, the solution to such SDE itself is a stable process and can be derived easily by least square parameter fitting from the observed flow concentration data. However, in a more generalized scenario, a closed form for the solution to a stable SDE may not exist. We propose a numerical method for solving/generating a stable SDE in a general set-up. The method incorporates a discretized finite volume scheme with the characteristic line to solve the fADE or the forward equation for the Markov process that solves the stable SDE. Then we use a numerical scheme to generate the solution to the governing SDE using the fADE solution. Also, often the functional form of the advection or dispersion coefficients are not known for a given plume concentration data to start with. We use a Levenberg–Marquardt (L-M) regularization method to estimate advection and dispersion coefficient function from the observed data (we present the case for a linear advection) and proceed with the SDE solution construction described above.