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Stability of quasi-stationary measures in high-dimensional products of mixing Markov chains — arXiv:2609.10178

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.10178
License (as archived): CC0-1.0
Archived: 2026-10-16T09:51:46.271Z
SHA-256 of the archived text: b33317efaac48b7f20c3e9417786d31534b33a7003db24ff9964c36132be1eb7
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Stability of quasi-stationary measures in high-dimensional products of mixing Markov chains — arXiv:2609.10178 — arXiv (en), https://arxiv.org/abs/2609.10178 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_b33ab0aaa595 (Metraipolis archive dc_b33ab0aaa595, sha256 b33317efaac48b7f20c3e9417786d31534b33a7003db24ff9964c36132be1eb7)

BibTeX

@misc{dcb33ab0aaa595,
  title = {Stability of quasi-stationary measures in high-dimensional products of mixing Markov chains — arXiv:2609.10178},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.10178},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_b33ab0aaa595; sha256 b33317efaac48b7f20c3e9417786d31534b33a7003db24ff9964c36132be1eb7},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.10178. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study high-dimensional conditioned dynamics obtained from an arbitrary number of independent copies of a mixing Markov chain. The dynamics is conditioned to avoid a family of holes whose stationary measure vanishes as the dimension grows, and we ask whether the resulting process admits a quasi-stationary measure close to the stationary product measure. Our problem is motivated by conditioning on holes with complicated geometry in high dimension, such as sets arising naturally from large deviations of suitable observables. Our main tool is the ANOVA decomposition, which separates functions according to their dependence on different subsets of coordinates. This allows us to exploit the product structure of the dynamics and obtain contraction estimates that are uniform in the dimension. Combined with a Keller Liverani perturbation argument, these estimates yield the existence of quasi-stationary densities converging to the stationary density as the size of the hole vanishes. Under stronger one-step mixing and regularization assumptions, we obtain sharper convergence in a Sobolev norm, with a square-root dependence on the measure of the hole. The improvement relies on the increasingly strong contraction of higher-order ANOVA components, thereby overcoming the usual loss of control with dimension. Finally, we apply our results to additive-noise Markov chains on the circle with smooth, uniformly positive transition densities

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-10 in math.PR).
Categorie arXiv: Dynamical Systems (math.DS) ; Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2609.10178 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T08:44:35.310Z dalla pagina https://arxiv.org/catchup/math.PR/2026-09-10?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).