Survival in a partially reactive wedge

Abstract

We investigate the power-law decay of the survival probability of a Brownian particle diffusing in an infinite planar wedge whose two sides are partially reactive and described by Robin boundary conditions. We employ matched asymptotic analysis to relate the long-time asymptotics to a stationary harmonic Robin problem near the apex. This approach determines the persistence exponent for arbitrary opening angles and yields the prefactor explicitly for the family of wedges with α=π/n (n=1,2,…). A simple extension of the apex prefactor to arbitrary angles is conjectured and supported numerically. This asymptotic behavior describes the crossover to the well-known result for perfectly absorbing wedges. As immediate applications, we also deduce the long-time behavior of the probability density function of the boundary local time on wedge sides, as well as the probability density function of the associated first-crossing time.

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BibTeX RIS

BibTeX

@misc{grebenkov2026survival,
  title = {{Survival in a partially reactive wedge}},
  author = {Denis S. Grebenkov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02665},
  eprint = {2609.02665},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02665},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Survival in a partially reactive wedge
AU  - Denis S. Grebenkov
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02665
UR  - https://arxiv.org/abs/2609.02665
AB  - We investigate the power-law decay of the survival probability of a Brownian particle diffusing in an infinite planar wedge whose two sides are partially reactive and described by Robin boundary conditions. We employ matched asymptotic analysis to relate the long-time asymptotics to a stationary harmonic Robin problem near the apex. This approach determines the persistence exponent for arbitrary opening angles and yields the prefactor explicitly for the family of wedges with α=π/n (n=1,2,…). A simple extension of the apex prefactor to arbitrary angles is conjectured and supported numerically. This asymptotic behavior describes the crossover to the well-known result for perfectly absorbing wedges. As immediate applications, we also deduce the long-time behavior of the probability density function of the boundary local time on wedge sides, as well as the probability density function of the associated first-crossing time.
ER  -

Abstract

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