A compounded random walk for space-fractional diffusion on finite domains

TL;DR

The paper addresses the challenge of defining space-fractional diffusion on finite domains by linking a compounded CTRW to a spectral fractional Laplacian. It introduces a one-dimensional compounded CTRW on $[a,b]$ with exponential waiting times and a $K$-step jump distribution governed by the Sibuya PGF $G_K(z)=1-(1-z)^{\alpha}$, yielding a diffusion-limit operator with eigenvalues $\lambda_n^{\alpha}$. In the diffusion limit the dynamics reduce to the space-fractional Fokker-Planck equation $\partial_t \rho = -D_{\alpha} (-\mathcal{L})^{\alpha} \rho$ with eigenfunctions $\chi_n$ and Robin boundary conditions, providing a well-defined operator on bounded domains. The framework is validated by Monte Carlo simulations that agree with analytical and spectral solutions and clarifies key differences from Lévy-flight based models, offering a practical route to modeling transport under space-dependent forces in finite regions.

Abstract

We formulate a compounded random walk that is physically well defined on both finite and infinite domains, and samples space-dependent forces throughout jumps. The governing evolution equation for the walk limits to a space-fractional Fokker-Planck equation valid on bounded domains, and recovers the well known superdiffusive space-fractional diffusion equation on infinite domains. We describe methods for numerical approximation and Monte Carlo simulations and demonstrate excellent correspondence with analytical solutions. This compounded random walk, and its associated fractional Fokker-Planck equation, provides a major advance for modeling space-fractional diffusion through potential fields and on finite domains.

Figures (5)

• Figure 1: A realization of a single particle performing a compounded CTRW where $\Delta x = 1, a = 0, b = 5$. The top panel shows the lattice site occupancy of the particle where the (orange) horizontal lines show positions and times when the particle is waiting and the (blue) vertical lines show positions sampled by steps during each instantaneous jump. The bottom panels show the $K$ random compounding steps performed by the random walker during jumps at times, $t_3$, $t_4$ and $t_8$.
• Figure 2: Solutions to the spectral fractional Fokker-Planck equation \ref{['eq:SpectralFractionalFokkerPlanckEquation']} (solid blue line) and the fractional advection dispersion equation \ref{['eq:jespersenffpe']} (orange dashed line) wtih $D_\alpha = 1$, $v = 1$, $\alpha =0.7$ at time $T = 1$, $2$, $4$, $8$.
• Figure 3: Comparison of solutions between the Monte Carlo simulations (orange), the numerical approximation (purple dots) and the first $1001$ terms of the analytical solution \ref{['eq:spectral_unbiased']} (black line) for the space-fractional diffusion equation on bounded domain $[-1,1]$ with reflecting boundaries. The parameters are $\alpha = 0.7$, $N = 10^6$ particles, $\gamma = 10^2$, $m = 301$ and $T = 5$. A subset of data points for the numerical approximation is shown for visual clarity.
• Figure 4: Comparison of solutions between the Monte Carlo simulations (orange), the numerical solution (purple dots) and the first $1001$ terms of the analytical solution \ref{['eq:spectral_biased']} (black line) for the space-fractional Fokker-Planck equation on bounded domain $[-1,1]$ with reflecting boundaries, where $\alpha = 0.7$, $N = 10^5$, $\gamma = 50$, $m = 201$ at multiple times, $T = 1$, $2$, $5$ and $10$.
• Figure 5: Monte Carlo simulations with $10^5$ particles (orange bars) and their corresponding numerical solutions (magenta dots) of the compounded CTRW in a finite domain $[-1,1]$ at times $T = 1$ (upper panels) and $2$ (lower panels). The case where particles are killed if any point on their path reaches the boundary is on the left and the case where particles are killed if the compounded jump process ends are outside the domain is on the right. The solution to the spectral fractional diffusion equation \ref{['eq:SolDir']} is shown as the same solid line on both the left and right panels. Here, $\alpha = 0.5$, the rate $\gamma = 50$ and the number of nodes is $m = 171$.