Green's functions of the fractional Laplacian on a square -- boundary considerations and applications to the Lévy flight narrow capture problem

TL;DR

This work addresses the nonlocal, boundary-sensitive narrow capture problem for Lévy flights in a 2D square by developing a Neumann-type fractional Laplacian operator and a robust, source-neutral Green's-function computation framework. The authors decompose the Green's function into an analytically known singular part and a smooth remainder, enabling accurate extraction of the regular part and its gradient without relying on explicit eigenfunction expansions. They derive two-term asymptotic expansions for the global mean first passage time and for splitting probabilities, illustrating boundary and shielding effects and revealing how Lévy flights cope with obstacles differently from Brownian motion. The methodology, applicable to periodic lattices and extendable to higher dimensions, provides precise quantitative tools for predicting search times and transition probabilities in nonlocal diffusion processes with boundaries and small targets.

Abstract

On the unit square, we introduce a method for accurately computing source-neutral Green's functions of the fractional Laplacian operator with either periodic or homogeneous Neumann boundary conditions. This method involves analytically constructing the singular behavior of the Green's function in a neighborhood around the location of the singularity, and then formulating a ``smooth'' problem for the remainder term. This smooth problem can be solved for numerically using a basic finite difference scheme. This approach allows accurate extraction of the regular part of the Green's function (and its gradient, if so desired). This new tool enables quantification of properties and characteristics of the narrow capture problem, where a particle undergoing a Lévy flight of index $α\in (0,1)$ searches for small target(s) of radius $\mathcal{O}(\varepsilon)$ for $0 < \varepsilon \ll 1$ on a bounded two-dimensional domain. In particular, it allows us to show how boundary interactions and configuration of multiple targets impact expected search time. Furthermore, we are able to illustrate how a target can be ``shielded'' by obstacles, and how a Lévy flight search can be significantly superior in navigating these obstacles versus Brownian motion. All asymptotic predictions are confirmed by full numerical solutions.

Figures (8)

• Figure 1: (a) In $\Omega$ with periodic boundary conditions, we center one circular target at $(0.25, 0.25)$ and another at $(s,s)$, where $0.25<s<0.75$. Both targets have radius $0<\varepsilon \ll 1$. (b) The global mean first passage time versus $s$ for $\alpha = 0.6$ and $\varepsilon = 0.03$. The red curve is generated from a finite difference solution for $u_\varepsilon^{(p)}$ satisfying \ref{['narrowescapeequationmult']} with $\mathcal{A}$ replaced by $\mathcal{A}_\alpha^{(p)}$, while the blue curve is obtained through an asymptotic analysis along with the algorithm for accurate computation of $G_\alpha^{(p)}$.
• Figure 2: (a) In $\Omega$ with reflective boundary conditions, we center a single circular target of radius $0<\varepsilon \ll 1$ at $(s,s)$, with $s \in (0,0.5)$. (b) The global mean first passage time versus $s$ for $\alpha = 0.6$ and $\varepsilon = 0.03$. The red curve is generated from a finite difference solution for $u_\varepsilon^{(n)}$ satisfying \ref{['narrowescapeequationmult']} with $\mathcal{A}$ replaced by $\mathcal{A}_\alpha^{(n)}$, while the blue curve is obtained through an asymptotic analysis along with the algorithm for accurate computation of $G_\alpha^{(n)}$.
• Figure 3: (a) In $\Omega$ with reflective boundary conditions, the desired target at the center (heavy line) is "shielded" by five obstacle targets. The average splitting probability, $\bar{v}_\varepsilon^{(n)}$, is the probability of reaching the desired target before hitting any of the obstacle targets, averaged over all starting locations in $\Omega$. (b) Plot of $\bar{v}_\varepsilon^{(n)}$ versus $\alpha$, where $\alpha = 1$ is the Brownian limit. The red curve is generated from a finite difference solution for $v_\varepsilon^{(n)}$ satisfying \ref{['splitting']}, while the blue curve is obtained through an asymptotic analysis along with the algorithm for accurate computation of $G_\alpha^{(n)}$. As expected, Lévy flights with smaller index $\alpha$, which experience more long jumps, are less susceptible to the shielding effect.
• Figure 4: Periodic boundary (a) versus specular reflection at the boundary (b). In (a), a particle whose trajectory causes it to exit the domain through the bottom boundary simply re-enters at the same angle from the top boundary. In (b), the same trajectory is reflected off the boundary at the same angle as the incident trajectory. The total distance traveled by the particle is the same in both cases; the distribution of this distance follows a power-law distribution.
• Figure 5: Four different paths from $\mathbf{x}$ to $\mathbf{y}$ in $\Omega = [0,1]\times[0,1]$ with reflective boundaries. The direct path from $\mathbf{x}$ to $\mathbf{y}$ of length $|\mathbf{y}-\mathbf{x}|$ has probability $\sim|\mathbf{y}-\mathbf{x}|^{-2-2\alpha}$. The other paths involve one or more reflections off of $\partial\Omega$, and have probability $\sim|T_{\bm}(\mathbf{y})~-~\mathbf{x}|^{-2-2\alpha}$ for $\bm \in \mathbb{Z}^2$.