An efficient probabilistic scheme for the exit time probability of $α$-stable Lévy process

TL;DR

The paper tackles the problem of computing exit time probabilities for symmetric $\alpha$-stable Lévy processes, which model anomalous diffusion with nonlocal jumps. It introduces a Gaussian approximation that represents the Lévy process as a Brownian component plus a finite-activity compound Poisson process, enabling a tractable PIDE framework for the exit problem via the Feynman-Kac formula. A fully discrete, quadrature-based scheme is developed to evaluate conditional expectations, with careful treatment of exit events and spatial interpolation, achieving first-order convergence in time and bypassing large dense linear systems. Numerical tests in 1D and 2D demonstrate accurate reproduction of the heavy-tailed, superdiffusive behavior and efficient computation of exit-time probabilities across a range of $\alpha$ and diffusivity regimes, including anisotropic transport settings.

Abstract

The α-stable Lévy process, commonly used to describe Lévy flight, is characterized by discontinuous jumps and is widely used to model anomalous transport phenomena. In this study, we investigate the associated exit problem and propose a method to compute the exit time probability, which quantifies the likelihood that a trajectory starting from an initial condition exits a bounded region in phase space within a given time. This estimation plays a key role in understanding anomalous diffusion behavior. The proposed method approximates the α-stable process by combining a Brownian motion with a compound Poisson process. The exit time probability is then modeled using a framework based on partial integro-differential equations (PIDEs). The Feynman-Kac formula provides a probabilistic representation of the solution, involving conditional expectations over stochastic differential equations. These expectations are computed via tailored quadrature rules and interpolation techniques. The proposed method achieves first-order convergence in time and offers significant computational advantages over standard Monte Carlo and deterministic approaches. In particular, it avoids assembling and solving large dense linear systems, resulting in improved efficiency. We demonstrate the method's accuracy and performance through two numerical examples, highlighting its applicability to physical transport problems.

Figures (8)

• Figure 1: The logarithmic probability density functions (log-pdf) of two processes, $L_t^{\alpha}$ and its approximation $\widehat{L}_t^{\alpha}$, at the unit time $t = 1$. The study tests three distinct $\alpha$ values: $1.75$, $1.5$, and $1.25$, each presented in a separate row. The blue curve represents the theoretical log-pdf produced through the MATLAB STBL toolbox stable, while the red curve depicts the log-pdf based on $N = 10^5$ samples. Both processes, $L_t^{\alpha}$ and $\widehat{L}_t^{\alpha}$, exhibit accurate adherence to the $\alpha$-stable distribution, as highlighted in the figure.
• Figure 2: This figure presents the mean-squared displacement (MSD) plots for the $S\alpha S$ process $L_t^{\alpha}$ and its Gaussian approximation $\widehat{L}_t^{\alpha}$, with separate panels corresponding to $\alpha = 1.75$, $1.5$, and $1.25$. The horizontal axis shows $\log(t)$, and the vertical axis shows $\log(\langle r^2 \rangle)$. Both processes exhibit the expected superdiffusive scaling behavior, with MSD growing as $t^{2/\alpha}$. The approximation $\widehat{L}_t^{\alpha}$ displays reduced fluctuations due to jump truncation but retains the same scaling rate, confirming that it preserves the key statistical feature of anomalous diffusion.
• Figure 3: Left panel: A comparison between $P_X(t,x)$ and $P_{\widehat{X}}(t,x)$ at $t=1$, with both values are computed using the direct Monte Carlo method (DMC). This plot demonstrates $P_{\widehat{X}}(t,x)$ serves as a reliable approximation for $P_X(t,x)$. Right panel: A comparison between DMC (ground truth) and BMC (our method) in solving $P_{\widehat{X}}(t,x)$. A good agreement between our method and ground truth.
• Figure 4: The exit time probability $P_{\widehat{X}}(t,x)$ at $t=1$ as a function of the diffusion coefficient $\chi$. The left panel corresponds to the initial position $x=0.5$, and the right panel to $x=0.1$. We observe a shift from local to nonlocal diffusion behavior as $\chi$ decreases. Additionally, near-boundary initial positions exhibit faster exits under local diffusion conditions.
• Figure 5: Error decay in the computation of $P_{\widehat{X}}(t,x)$ at $t=1$ for $\alpha = 1.0$, $1.25$, $1.5$ and $1.75$. The error is measured by the $l^2$-norm of the difference between BMC result and the ground truth (computed by DMC). The proposed method achieves the first-order convergence rate in $\Delta t$.