Diffusion with stochastic resetting in the presence of a delta killing trap: Drift-controlled survival regimes
Abstract
We study a one-dimensional diffusive particle subject to stochastic resetting to its initial position, in the presence of an imperfect, localized target that can absorb (kill) the particle, modeled by a delta-function killing rate. The central question is how stochastic resetting competes with drift-induced transience and the target's finite reactivity. Exact Laplace-space expressions are derived for the non-normalized propagator and survival probability, for arbitrary diffusion coefficient D, resetting rate r, and killing strength k, first in the absence of drift and then under a constant drift. The long-time behavior separates into distinct regimes. Without resetting, unbiased diffusion exhibits the recurrent algebraic survival law S(t)∼ t−1/2, whereas any nonzero drift renders the motion transient with respect to the target and leaves a nonzero ultimate survival probability. By contrast, any r>0 and k>0 repeatedly renews encounters with the target and restores eventual absorption, yielding an exponential survival law S(t)∼ A e−θt. The decay rate θ is given by the dominant (rightmost) pole of the Laplace transform, and the surviving density converges after normalization to an explicit quasi-stationary profile. The driftless and perfectly absorbing limits recover standard resetting results. These formulas provide a unified description of the crossover between recurrence-controlled, transience-controlled, and renewal-controlled survival.
Versions
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BibTeX
@misc{mazzolo2026diffusion,
title = {{Diffusion with stochastic resetting in the presence of a delta killing trap: Drift-controlled survival regimes}},
author = {Alain Mazzolo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22527},
eprint = {2609.22527},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22527},
}
RIS
TY - UNPB
TI - Diffusion with stochastic resetting in the presence of a delta killing trap: Drift-controlled survival regimes
AU - Alain Mazzolo
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22527
UR - https://arxiv.org/abs/2609.22527
AB - We study a one-dimensional diffusive particle subject to stochastic resetting to its initial position, in the presence of an imperfect, localized target that can absorb (kill) the particle, modeled by a delta-function killing rate. The central question is how stochastic resetting competes with drift-induced transience and the target's finite reactivity. Exact Laplace-space expressions are derived for the non-normalized propagator and survival probability, for arbitrary diffusion coefficient D, resetting rate r, and killing strength k, first in the absence of drift and then under a constant drift. The long-time behavior separates into distinct regimes. Without resetting, unbiased diffusion exhibits the recurrent algebraic survival law S(t)∼ t^−1/2, whereas any nonzero drift renders the motion transient with respect to the target and leaves a nonzero ultimate survival probability. By contrast, any r>0 and k>0 repeatedly renews encounters with the target and restores eventual absorption, yielding an exponential survival law S(t)∼ A e^−θt. The decay rate θ is given by the dominant (rightmost) pole of the Laplace transform, and the surviving density converges after normalization to an explicit quasi-stationary profile. The driftless and perfectly absorbing limits recover standard resetting results. These formulas provide a unified description of the crossover between recurrence-controlled, transience-controlled, and renewal-controlled survival.
ER -