[Paper Review] Dynamic redundancy and mortality in stochastic search

Samantha Linn, Aanjaneya Kumar|arXiv (Cornell University)|Jan 11, 2026

Diffusion and Search Dynamics0 citations

TL;DR

This paper develops a general framework (dynamic redundancy and mortality, DRM) for stochastic search with ongoing recruitment and death, derives exact first-passage time statistics, and compares DRM to stochastic resetting, with a detailed Brownian 1D case.

ABSTRACT

Search processes are a fundamental part of natural and artificial systems. In such settings, the number of searchers is rarely constant: new agents may be recruited while others can abandon the search. Despite the ubiquity of these dynamics, their combined influence on search efficiency remains unexplored. Here we present a general framework for stochastic search in which independent agents progressively join and leave the process, a mechanism we term dynamic redundancy and mortality (DRM). Under minimal assumptions on the underlying search dynamics, this framework yields exact first-passage time statistics. It further reveals surprising connections to stochastic resetting, including a regime in which the resetting mean first-passage time emerges as a universal lower bound for DRM, as well as regimes in which DRM search is faster. We illustrate our results through a detailed analysis of one-dimensional Brownian DRM search. Altogether, this work provides a rigorous foundation for studying first-passage processes with a fluctuating number of searchers, with direct relevance across physical, biological, and algorithmic systems.

Motivation & Objective

• Motivate stochastic search scenarios where searcher populations fluctuate due to recruitment and mortality.
• Introduce the DRM framework and establish exact first-passage time statistics under minimal assumptions.
• Express the DRM survival probability in terms of single-mortal searcher statistics and derive universal MFPT bounds.
• Relate DRM to stochastic resetting and identify regimes where DRM outperforms resetting.
• Provide a detailed analysis of Brownian DRM search in one dimension to illustrate the theory.

Proposed method

• Define DRM survival probability S_{λ,μ}(t) and relate it to single-mortal survival S_{0,μ}(t) via S_{λ,μ}(t) = S_{0,μ}(t) exp{−λ ∫_0^t [1−S_{0,μ}(t')] dt'}.
• Express S_{0,μ}(t) in terms of the single-searcher FPT density P(τ=t) as S_{0,μ}(t) = 1 − ∫_0^t e^{−μ t'} P(τ=t') dt'.
• Obtain the mean first-passage time E[T_{λ,μ}] from S_{λ,μ}(t) via E[T_{λ,μ}] = ∫_0^∞ S_{λ,μ}(t) dt.
• Derive universal bounds on E[T_{λ,μ}] in terms of p_{μ} = ∫_0^∞ e^{−μ t} P(τ=t) dt, yielding lower and upper bounds.
• Show a connection to stochastic resetting; in particular, balanced DRM (λ=μ) has resetting MFPT as a universal lower bound.
• Specialize to Brownian motion in 1D to illustrate explicit forms and optimal turnover effects.

Experimental results

Research questions

• RQ1 What is the survival probability S_{λ,μ}(t) and the MFPT E[T_{λ,μ}] for DRM when searchers dynamically join and leave the process?
• RQ2 What universal bounds constrain the DRM MFPT, and how do they depend on the single-searcher mortality rate μ and recruitment rate λ?
• RQ3 How does DRM compare to stochastic resetting, especially in the balanced case λ=μ and in high turnover regimes?
• RQ4 Under Brownian motion in one dimension, what specific bounds and optimal parameter regimes arise for DRM?

Key findings

• DRM yields an exact expression for the survival probability in terms of single-mortal searcher statistics: S_{λ,μ}(t) = S_{0,μ}(t) exp{−λ ∫_0^t (1−S_{0,μ}(t′)) dt′}.
• MFPT under DRM is finite for all positive λ and μ and satisfies universal bounds: (1−p_{μ})/(λ p_{μ}) ≤ E[T_{λ,μ}] ≤ (1−p_{μ})/(λ p_{μ}) − ∂_{μ} log p_{μ}, with p_{μ} = ∫_0^∞ e^{−μ t} P(τ=t) dt.
• In the balanced case (λ = μ), stochastic resetting provides a universal lower bound for DRM MFPT.
• DRM can outperform stochastic resetting when recruitment dominates mortality (sufficient turnover), with explicit results in the 1D Brownian case showing regimes where DRM MFPT is smaller than resetting MFPT.
• In Brownian 1D, explicit bounds (e.g., E[T_{λ,μ}] between expressions involving x0, D, and √μ) and an optimal turnover rate exist, with a phase diagram distinguishing DRM- and resetting-dominated regions.
• The framework connects DRM to resetting on macroscopic scales while revealing distinct trajectory-level differences in FPT statistics.