Monte Carlo Approximations of Time-Nonlocal Diffusions in Bounded Domains

Abstract

We develop and analyze a Monte Carlo method for sampling killed anomalous diffusions obtained by time-changing Brownian motion with drift by the inverse of a subordinator. The method targets probabilistic representations of time-nonlocal, including time-fractional, Cauchy--Dirichlet problems on bounded domains. Since inverse subordinators can be sampled exactly in broad classes, while Brownian exit times are generally unavailable in arbitrary domains, we approximate the killed Brownian component by an Euler scheme with discrete boundary detection. We prove a square-root weak error bound with explicit dependence on the Laplace exponent of the subordinator, and derive mean-square and central limit results for the resulting Monte Carlo estimator. A numerical example in the disk and one in a high-dimensional anisotropic shell illustrate the theoretical rates, computation time, and the mesh-free character of the method.

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BibTeX

@misc{biocic2026monte,
  title = {{Monte Carlo Approximations of Time-Nonlocal Diffusions in Bounded Domains}},
  author = {Ivan Biočić and Daniel E. Cedeño-Girón and Aleksandar Mijatović and Bruno Toaldo},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20128},
  eprint = {2609.20128},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20128},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Monte Carlo Approximations of Time-Nonlocal Diffusions in Bounded Domains
AU  - Ivan Biočić
AU  - Daniel E. Cedeño-Girón
AU  - Aleksandar Mijatović
AU  - Bruno Toaldo
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20128
UR  - https://arxiv.org/abs/2609.20128
AB  - We develop and analyze a Monte Carlo method for sampling killed anomalous diffusions obtained by time-changing Brownian motion with drift by the inverse of a subordinator. The method targets probabilistic representations of time-nonlocal, including time-fractional, Cauchy--Dirichlet problems on bounded domains. Since inverse subordinators can be sampled exactly in broad classes, while Brownian exit times are generally unavailable in arbitrary domains, we approximate the killed Brownian component by an Euler scheme with discrete boundary detection. We prove a square-root weak error bound with explicit dependence on the Laplace exponent of the subordinator, and derive mean-square and central limit results for the resulting Monte Carlo estimator. A numerical example in the disk and one in a high-dimensional anisotropic shell illustrate the theoretical rates, computation time, and the mesh-free character of the method.
ER  -

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