First-Passage Times for the Space Fractional Fokker-Planck Equation

Abstract

We extend the random walk framework to include compounded steps, providing first-passage time (FPT) properties for a new class of superdiffusive processes, which are governed by the space-fractional spectral Fokker-Planck equation. This first-passage process leads to novel FPT properties, different from Lévy flights and walks, that account for space dependent forces and hitting boundaries throughout the path of a jump. The FPT distribution can be derived for different types of barriers and potentials, for which we also provide specific examples. For the one-sided absorbing boundary on the semi-infinite line, we find that the FPT density scales as $t^{-1/(2α)-1}$, in agreement with the method of images but in violation of the Sparre-Andersen scaling. In this case, there exists an optimal space-fractional exponent $α$ to minimize the mean FPT.

Figures (5)

• Figure 1: Example realization of a discrete time random walk up to $n^*$ (top) where each step corresponds to a step in continuous time (bottom). Here, the domain is $\Omega = \{0,1,2,3,4,5\}$ with $\Omega' = \{0\}$ as the target region. The number of compounded steps in each interval are $K_1 = 3$, $K_2 = 8$, $K_3 = 1$ and $K_4 = 4$ but is absorbed before completing the $4$ steps between times $(3\Delta t,4\Delta t]$.
• Figure 2: The PDF \ref{['Unbiased2s_rho']} for finding the particle at $x$, where $0<x<L$ for $t = 0.5$ (top) and $1.0$ (bottom) truncated after the first $1000$ terms. Here, $D_{\alpha} = 1$, $L = 2$ (left) and $4$ (right), $x_0 = L/2$ for $\alpha = 0.4$, $0.6$ and $1.0$.
• Figure 3: Plot of the survival function \ref{['UnbiasedH_sur']} (top) and PDF \ref{['UnbiasedH_PDF']} (bottom) of the FPT for the one-sided absorbing diffusion process on the semi-infinite interval approximated with numerical integration. Here, $D_\alpha = 1$, $x_0 = 2$, $0 \leq t \leq 5$ for $\alpha = 0.2$, $0.4$, $0.6$, $0.8$ and $1.0$.
• Figure 4: Log-log plot for the PDF \ref{['UnbiasedH_PDF']} (solid lines) and its asymptotic behavior $f(t,\alpha)$\ref{['asym']} (dashed black lines) of the FPT for the one-sided absorbing diffusion process on the semi-infinite interval for $\alpha = 0.4$, $0.6$, $0.8$ and $1.0$ starting at $x_0 = 2$. The corresponding plot for the FPT Monte Carlo results of the underlying compounded random walk \ref{['eq:embedding']} (solid filled bars) and Lévy flight (dotted lines) for $10^6$ particles. Here, $\Delta t = 0.125$ and $\Delta x$ was determined by setting $D_\alpha = 1$ and using \ref{['eq:D_alpha']} for $\alpha = 0.4$, $0.6$, $0.8$ and $1.0$, respectively. To compare the FPT PDF of Lévy flights to the scaling $\propto t^{-3/2}$ we have included the red dotted-dashed lines in each subplot except for $\alpha = 1$, since $f(t,1.0) \propto t^{-3/2}$.
• Figure 5: The expected FPT for the one-sided absorbing diffusion process on the semi-infinite interval \ref{['UnbiasedH_exp']}. Here, $D_\alpha = 1$ and $0.0 < \alpha < 0.5$ for $x_0 = 0.04$, $0.20$, $1.00$, $5.00$ and $25.00$.