Annealed Survival Probability of Random Walk in an Inhomogeneous Poisson Environment of Mobile Traps

Technical Summary: Annealed Survival Probability of Random Walk in an Inhomogeneous Poisson Environment of Mobile Traps

Problem Statement
This paper investigates the annealed survival probability of a continuous-time simple symmetric random walk ($X$) moving on the integer lattice $\mathbb{Z}^d$ in the presence of mobile traps. Unlike previous studies focusing on homogeneous environments, this work considers an inhomogeneous Poisson environment. Specifically, at time $t=0$, the number of traps at each site $y \in \mathbb{Z}^d$ is distributed as a Poisson random variable with mean $\nu_y$, where the sequence $(\nu_y)_{y \in \mathbb{Z}^d}$ is not necessarily constant. Each trap performs an independent simple symmetric random walk with jump rate 1. The walker $X$ (with jump rate $\kappa \ge 0$) is annihilated at a rate $\gamma$ proportional to the number of traps at its current location.

The primary objective is to determine the asymptotic behavior of the annealed survival probability $Z_t^\gamma$ (the expectation over both the random walk and the trap field) as time $t \to \infty$, specifically in dimensions $d=1$ and $d=2$. Additionally, the paper establishes limit theorems (Law of Large Numbers, Central Limit Theorem, and Large Deviation Principles) for occupational time functionals of the trap environment at the origin.

Methodology and Assumptions
The analysis relies on specific assumptions regarding the initial trap intensity profile $(\nu_y)$:

• (A1) The initial means are uniformly bounded: $\sup_y \nu_y < \infty$.
• (A2) The expected trap density at the origin is minimal for all times: $m(t, x) \ge m(t, 0)$ for all $x \in \mathbb{Z}^d, t \ge 0$, where $m(t, x)$ is the mean number of traps at $x$ at time $t$.
• (A3) A weaker condition on the time-averaged density at the origin, allowing for polynomial or logarithmic scaling, defined via a slowly varying function $L(t)$ and exponent $\alpha$.

The methodology diverges from the homogeneous case (e.g., [DGRS12]) due to the lack of time-reversibility in the inhomogeneous trap field, which prevents the direct application of the Feynman-Kac representation and subadditivity arguments used to prove the existence of a Lyapunov exponent. Instead, the author employs:

1. Cumulant Expansion: Adapted from [CG84] to handle the inhomogeneous Poisson field, deriving integral identities for the survival probability conditioned on the walker's trajectory.
2. Pascal Principle: A continuous-time proof showing that, under assumption (A2), the survival probability is maximized when the walker remains stationary at the origin ($X \equiv 0$). This provides the necessary upper bound.
3. Confinement Strategy: A lower bound is derived by confining the walker to a ball of radius $R_t$ where no traps are initially present and no external traps enter during time $t$.
4. Tauberian Theorems and Local Limit Theorems: Used to analyze the asymptotic behavior of Laplace transforms of the relevant functionals to derive precise decay rates.

Key Results

1. Asymptotics of Annealed Survival Probability (Theorem 1.1):
   Under assumptions (A1) and (A2), the paper establishes precise asymptotics for $Z_t^\gamma$ in dimensions $d=1$ and $d=2$. The leading-order decay is independent of the killing rate $\gamma$ and depends on the time-averaged trap density at the origin, $\bar{m} = \lim_{t\to\infty} \frac{1}{t} \int_0^t m(s, 0) ds$.

   • $d=1$: $Z_t^\gamma \sim \exp\left( -\bar{m} \sqrt{\frac{8}{\pi}} t^{1/2} \right)$.
   • $d=2$: $Z_t^\gamma \sim \exp\left( -\pi \bar{m} \frac{t}{\log t} \right)$.
     These results generalize the homogeneous findings of [DGRS12], showing that the decay rate is determined by the average trap intensity at the origin rather than the global spatial average, provided the density is minimal at the origin.

2. Limit Theorems for Occupational Functionals (Theorems 1.2 & 1.3):
   The paper proves limit theorems for two functionals:

   • $D_t(\xi)$: The total time traps spend at the origin.
   • $N_t(\xi)$: The number of distinct traps that visit the origin.
     Under the weaker assumption (A3), the paper establishes:
   • Law of Large Numbers: Both functionals converge almost surely to their expectations.
   • Central Limit Theorem: The fluctuations are Gaussian with variance depending on dimension and $\bar{m}$.
   • Large Deviation Principle: The probability of deviations decays exponentially with a rate function $I(\beta)$ explicitly calculated for each dimension.
     These results generalize the homogeneous case results from [CG84].

3. Examples and Counter-Examples (Section 1.2):

   • Periodic and Decaying Profiles: Theorems apply to periodic environments (Example 1.1) and profiles where $\nu_y \to 1$ as $|y| \to \infty$ (Example 1.2), recovering the standard decay rates with $\bar{m}$ as the appropriate average.
   • Failure of (A2): When the trap density is not minimal at the origin (e.g., $\nu_y$ decays as $|y|^{-\beta}$ with $0 < \beta < 1$), the survival probability decays strictly slower than the homogeneous case (Example 1.3). In cases where the total initial trap mass is finite (Example 1.4), the survival probability does not decay to zero at all.

Significance and Claims
The paper claims to extend the rigorous understanding of trapping problems from homogeneous to inhomogeneous settings. Its primary significance lies in:

• Overcoming Non-Reversibility: It addresses the technical challenge that inhomogeneous trap fields are not time-reversible, rendering standard Feynman-Kac and subadditivity techniques inapplicable. The author successfully adapts cumulant expansion and proves a continuous-time Pascal principle for inhomogeneous fields.
• Characterizing Inhomogeneity: It demonstrates that the asymptotic decay rate is governed by the local behavior of the trap intensity at the origin (specifically, the time-averaged minimum density), rather than the global distribution, provided the density is minimized at the origin.
• Generalization: It unifies and generalizes the results of [DGRS12] (homogeneous mobile traps) and [CG84] (homogeneous occupational times) to a broader class of initial configurations.

The author notes that the problem remains open for dimensions $d \ge 3$ due to the transience of the random walk, which invalidates the confinement strategy used for lower bounds. Furthermore, while lower bounds are established for cases where assumption (A2) fails (but A3 holds), matching upper bounds are not proven, leaving the exact asymptotic rate in those specific inhomogeneous cases as an open conjecture.