Worst-case mixing estimates for Brownian motion with semipermeable barriers

TL;DR

The paper analyzes how semipermeable barriers affect mixing for Brownian motion confined to a planar domain, introducing the snapping-out Brownian motion model. It develops a Doeblin-minorization framework built from local transport steps to bound the probability flow between regions and then aggregates these steps along geodesics to obtain global mixing-time and stationary-density bounds expressed in terms of geometry and barrier permeability. A key finding is that, in worst-case configurations, the bounds decay exponentially with domain size, and explicit examples show this exponential decay is necessary. The results provide qualitative, geometry-driven guarantees applicable to Monte Carlo methods and barrier-recovery problems, highlighting how barrier arrangement and permeability asymmetry shape convergence behavior.

Abstract

We study the mixing properties of a Brownian motion whose movements are hindered by semipermeable barriers. Our setting assumes that the process takes values in a smooth planar domain and that the barriers are one-dimensional closed curves. We establish an upper bound on the mixing time and a lower bound on the stationary distribution in terms of geometric parameters. These worst-case bounds decay at an exponential rate as the domain grows large, and we give examples that show that exponential decay is necessary in our worst-case setting.

Figures (7)

• Figure 1: Examples of pathological geometries that could complicate the mixing properties. A bound on $\kappa$ allows ruling out the leftmost example, but is not sufficient to rule out the two examples on the right. This is where a bound on $\rho$ is required.
• Figure 2:
• Figure 3: Visualization of the event described in the probability of \ref{['eq:TediousFish']} from Lemma \ref{['lem: PillShaped']}. This event asks that the Wiener process $W_t$, represented by the dotted blue line, progresses in a straight line towards $y$ up to a moderate deviation of distance $\leq \gamma R$ and that $W_{R^2}$ ends up in a neighborhood of radius $\varepsilon$ of $y$. Here, $\varepsilon >0$ could be arbitrarily small.
• Figure 4: Visualization of Lemma \ref{['lem: ReductionSameSide']} and its proof idea. Using Lemma \ref{['lem: PillShaped']}, we can ensure that $X_t$ runs into the barrier with nontrivial probability. Then, also using that $\lambda_{\min}$ and $\lambda_{\max}$ control how quickly $s_i$ changes, we can estimate the probability that $X_t$ crosses the barrier exactly once. This allows proving that $X_{ r_\delta^2}$ lands in $\mathcal{Z}(\delta,y)$ with nontrivial probability.
• Figure 5: Visualization of the proof idea for Lemma \ref{['lem: TransportSameSide']}. By using Lemma \ref{['lem: PillShaped']} on a sufficiently narrow neighborhood of the line connecting $x_0$ to $y$, we can ensure that the manufactured process $x_0 + W_t$ never hits any barrier. Then, $X_{r_\delta^2} = x_0 + W_{r_\delta^2}$ lands in the ball of radius $\varepsilon$ around $y$.

Theorems & Definitions (42)

• Definition 1
• Theorem 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3: Lemma 3.5 in vanwerde2024recovering
• Lemma 3.4: Stay in ball if local time is controlled
• proof
• Lemma 3.5: Barrier is crossed at least once