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Diffusion under competing bulk and surface stopping mechanisms

  • Yilin Ye
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Abstract

We investigate reflected diffusion in a bounded domain subject to two independent, competing stopping mechanisms: an exponentially distributed bulk lifetime of rate p and a surface reaction triggered when the boundary local time exceeds an independent exponential threshold of rate q. Denoting by T the stopping time and by L the acquired boundary local time at stopping, we derive their marginal distributions, joint Laplace transform, and complete hierarchy of mixed moments. These statistics are determined by the splitting probability ϕ that surface reaction occurs before bulk decay. In particular, we establish the identity pexpectT+qexpectL=1 and show that the cumulative risk pT+qL is exponentially distributed with unit rate. We further obtain equivalent representations of ϕ in terms of the Robin-Laplacian and the generalized Steklov spectra. Explicit results for a three-dimensional ball reveal how competing rates p,q control ϕ and the (T,L) statistics. Monte Carlo simulations test the universal cumulative-risk law.

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Cite

BibTeXRIS
BibTeX
@misc{ye2026diffusion,
  title = {{Diffusion under competing bulk and surface stopping mechanisms}},
  author = {Yilin Ye},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05247},
  eprint = {2609.05247},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05247},
}
RIS
TY  - UNPB
TI  - Diffusion under competing bulk and surface stopping mechanisms
AU  - Yilin Ye
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05247
UR  - https://arxiv.org/abs/2609.05247
AB  - We investigate reflected diffusion in a bounded domain subject to two independent, competing stopping mechanisms: an exponentially distributed bulk lifetime of rate p and a surface reaction triggered when the boundary local time exceeds an independent exponential threshold of rate q. Denoting by T the stopping time and by L the acquired boundary local time at stopping, we derive their marginal distributions, joint Laplace transform, and complete hierarchy of mixed moments. These statistics are determined by the splitting probability ϕ that surface reaction occurs before bulk decay. In particular, we establish the identity pexpectT+qexpectL=1 and show that the cumulative risk pT+qL is exponentially distributed with unit rate. We further obtain equivalent representations of ϕ in terms of the Robin-Laplacian and the generalized Steklov spectra. Explicit results for a three-dimensional ball reveal how competing rates p,q control ϕ and the (T,L) statistics. Monte Carlo simulations test the universal cumulative-risk law.
ER  -