[Paper Review] On the Escape of a Random Walk From Two Pieces of a Tripartite Set

Michael Carlisle|arXiv (Cornell University)|Sep 8, 2012

Diffusion and Search Dynamics1 references3 citations

TL;DR

This paper derives sharp bounds for the Green's function and expected hitting times of a random walk on a tripartite state space $\Omega = A \sqcup B \sqcup C$, where $C$ separates $A$ and $B$. Using strong Markov property and excursion decomposition, it establishes upper and lower bounds for $G_{A\cup B}(x,y)$ and $\mathbb{E}^x[T_C]$, showing that escape from $A \cup B$ to $C$ is controlled by the probabilities of jumping between $A$ and $B$ before hitting $C$, with key bounds depending on $\psi_a = \mathbb{P}^a(T_B < T_C)$ and $\sigma_b = \mathbb{P}^b(T_A < T_C)$.

ABSTRACT

Let $\{A, B, C\}$ be a partition of a sample space $Ω$. For a random walk $S_n = x + \sum_{j=1}^n X_j$ starting at $x \in A$, we find estimates for the Green's function $G_{A \cup B}(x,y)$ and the hitting time $E^x(T_C)$ for $x, y \in A \cup B$, with interest in the case where $C$ "separates" $A$ and $B$ in a sense (e.g. the probability of jumping from $A$ to $B$, or vice versa, before hitting $C$, is small).

Motivation & Objective

• To analyze the escape behavior of a random walk from a union of two sets $A \cup B$ when a third set $C$ separates them.
• To derive quantitative bounds on the Green's function $G_{A\cup B}(x,y)$ for $x,y \in A \cup B$ in terms of hitting probabilities to $C$.
• To estimate the expected hitting time $\mathbb{E}^x[T_C]$ for $x \in A \cup B$, especially when transitions between $A$ and $B$ before hitting $C$ are rare.
• To quantify the difference between hitting distributions on $C$ and $C \cup A$ via the term $p(b,c,C,A) \leq \sigma_b = \mathbb{P}^b(T_A < T_C)$.

Proposed method

• Uses the strong Markov property at first hitting times $T_A$, $T_B$, and $T_C$ to decompose paths into excursions between $A$, $B$, and $C$.
• Applies the last exit decomposition to express hitting distributions $H_A(x,y)$ as sums over exit points from $A^c$, using Green's functions on $A^c$.
• Derives recursive bounds for $G_{A\cup B}(x,y)$ by conditioning on first passage to $B$ from $A$ and back, using $\rho_a = \mathbb{P}^a(T_B, T^*_A < T_C)$ and $\phi_b = \mathbb{P}^b(T_A, T^*_B < T_C)$.
• Establishes upper bounds for $\mathbb{E}^x[T_C]$ by decomposing the hitting time into stages: $T_{B\cup C}$, then $T_C$ after exiting to $B$, using $\psi_a = \mathbb{P}^a(T_B < T_C)$ and $\sigma_b = \mathbb{P}^b(T_A < T_C)$.
• Uses geometric series summation to solve the recurrence in expected hitting time bounds, yielding closed-form expressions involving $\psi\sigma$.
• Borrows and adapts classical results on Green's functions and hitting distributions, particularly from Spitzer and others, to the tripartite setting.

Experimental results

Research questions

• RQ1 How does the Green's function $G_{A\cup B}(x,y)$ for $x,y \in A \cup B$ behave when $C$ separates $A$ and $B$?
• RQ2 What are the tight upper and lower bounds for $\mathbb{E}^x[T_C]$ when the walk starts in $A \cup B$ and $C$ acts as a separator?
• RQ3 How does the hitting distribution on $C$ differ from that on $C \cup A$ when starting from $B$, and what controls this difference?
• RQ4 What role do the probabilities $\psi_a = \mathbb{P}^a(T_B < T_C)$ and $\sigma_b = \mathbb{P}^b(T_A < T_C)$ play in controlling escape from $A \cup B$?
• RQ5 Can the expected time to hit $C$ be bounded recursively using excursions between $A$ and $B$ before absorption in $C$?

Key findings

• The Green's function on $A \cup B$ satisfies $G_{A\cup B}(a,a') \leq G_A(a,a') + \frac{\rho_a}{1 - \rho_{a'}} G_A(a',a')$ for $a,a' \in A$, where $\rho_a = \mathbb{P}^a(T_B, T^*_A < T_C)$.
• For $x \in A$, the expected hitting time to $C$ satisfies $\mathbb{E}^x[T_C] \leq \mathbb{E}^x[T_{B\cup C}] + \psi_x \left[ \frac{f_B + \sigma f_A}{1 - \psi\sigma} \right]$, with $\psi_x = \mathbb{P}^x(T_B < T_C)$ and $\sigma = \sup_{b \in B} \mathbb{P}^b(T_A < T_C)$.
• The difference between the hitting distribution on $C$ and on $C \cup A$ from $b \in B$ is bounded by $p(b,c,C,A) \leq \sigma_b = \mathbb{P}^b(T_A < T_C)$, indicating that small $\sigma_b$ implies $H_C(b,c) \approx H_{C\cup A}(b,c)$.
• The bounds are sharp in the sense that they depend only on the escape probabilities from $A$ to $B$ and from $B$ to $A$ before hitting $C$, with convergence controlled by $\psi\sigma < 1$.
• The recurrence structure in the hitting time bounds leads to a geometric series with ratio $\psi\sigma$, ensuring convergence when the probability of multiple excursions is subcritical.
• The paper establishes that $G_{A\cup B}(a,b) \leq \min\left\{ \frac{\sigma_b}{1 - \rho_a} G_A(a,a), \frac{\psi_a}{1 - \phi_b} G_B(b,b) \right\}$, showing the off-diagonal Green's function is controlled by the probability of crossing between $A$ and $B$ before hitting $C$.