Diffusion in a wedge geometry: First-Passage Statistics under Stochastic Resetting

TL;DR

This work analyzes Brownian diffusion inside a 2D wedge with absorbing edges under stochastic resetting. It delivers exact and semi-analytic results for unconditional and conditional first-passage statistics, including the PDF, survival probability, boundary currents, and splitting probabilities, by employing renewal theory and Laplace transforms. A key contribution is the CV-based phase diagram that identifies parameter regimes where resetting speeds up mean absorption, including a closed-form critical wedge angle $\alpha_c^{\max}\approx0.36216$ rad and a universal criterion for conditioned exit outcomes. The findings show resetting can bias exit through the boundary closer to the reset point and can optimize conditioned and unconditioned first-passage times, with validation via Langevin-type simulations. These insights advance understanding of resetting in geometrically constrained diffusion and suggest strategies for targeted transport in wedge-like domains.

Abstract

We study the diffusion process in the presence of stochastic resetting inside a two-dimensional wedge of top angle $α$, bounded by two infinite absorbing edges. In the absence of resetting, the second moment of the first-passage time diverges for $α>π/4$ while it remains finite for $α<π/4$, resulting in an unbounded or bounded coefficient of variation in the respective angular regimes. Upon introducing stochastic resetting, we analyze the first-passage properties in both cases and identify the geometric configurations in which resetting consistently enhances the rate of absorption or escape through the boundaries. By deriving the expressions for the probability currents and conditional first-passage quantities such as splitting probabilities and conditional mean first-passage times, we demonstrate how resetting can be employed to bias the escape pathway through the favorable boundary. Our theoretical predictions are verified through Langevin-type numerical simulations, showing excellent agreement.

Figures (7)

• Figure 1: Schematic of a Brownian particle diffusing inside a two dimensional wedge. The particle starts from ($r_0,\theta_0$) and it is being reset intermittently at a rate $\lambda$ to the same location from where it renews the motion. The wedge has two infinite absorbing edges separated at an angle $\alpha$. In this study we analyze the first-passage time to any of these edges (e.g., facilitated by a trajectory in brown dashed) under resetting mechanism.
• Figure 2: Mean first-passage time $\left< t_\lambda \right >$ vs reset rate $\lambda$ showing optimal behaviour. The analytical expression for the MFPT [Eq. (\ref{['mfptwithreset']})] for two sets of parameters, dashed blue line ($r_0 = 2,~\alpha=\pi/3,~\theta_0=\pi/6,~D=1$) and solid light green line (inset) ($r_0 = 3,~\alpha=\pi/4,~\theta_0=\pi/6,~D=0.5$) are compared against numerical simulations (Black and red dots respectively), showing good agreement.
• Figure 3: The coefficient of variation (CV) plotted as a function of system parameters, with the vertical axis representing, wedge angle $\alpha$ and the horizontal axis representing the ratio $\Theta_0\equiv\theta_0/\alpha$ with $\theta_0$ being the initial angle ($0<\theta_0\le\alpha$). Three distinct regions are identified: In region I (depicted in orange)), $\text{CV} < 1$, indicating that resetting does not expedite the process for any $(\theta_0, \alpha)$ and in region II (in cyan), $\text{CV} > 1$, where resetting can optimize the MFPT. In region III (in purple), the second moment has no finite value while the first moment remains finite, causing the CV to diverge -- resetting is always beneficial in such cases with MFPT attaining a global minimum for $\lambda > 0$. The point(Black asterisk) corresponding to the maximum of the critical wedge angle $\alpha_c^{max}$ has also been represented in the figure. The colored asterisk symbols(Blue, Green and Red) in each region represents parameter values for which MFPT is plotted as a function of $\lambda$ (in the inset) corroborating the above findings.
• Figure 4: Plot of $\left<t_\lambda\right>_{\pm}$ [Eq. (\ref{['cmfpttimedomain']})] and $\left<t_\lambda\right>$ [Eq. (\ref{['mfptwithreset']})] plotted as a function of the reset angle $\theta_0$, evaluated for the parameters, $\alpha=\frac{\pi}{3},~\lambda=0.3,~D=1~\text{and}~r_0=2.$
• Figure 5: Plot showing conditional mean first-passage times (CMFPT) versus reset rate $\lambda$, for both asymmetric ($\theta_0 \ne \alpha/2$) and for symmetric $\theta_0 = \alpha/2$ (inset) resetting from the analytical expression given in Eq. (\ref{['cmfpttimedomain']}). The parameters are, $\alpha = \pi/3,~r_0 =2,~D=1$, with $\theta_0=\pi/5$ (main plot) and $\theta_0=\pi/6$ (inset). CMFPT is seen to attain an optimal value for the parameter values. Black squares and gray dots respectively in the main and inset plots are data points obtained from numerical simulations.