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PreprintarXiv licencearXiv

Posterior Consistency for Recovering Initial States in Nonlinear Subdiffusion Equations

  • Haoyu Lu
  • Shaokang Zu
  • Junxiong Jia
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Abstract

We study the Bayesian recovery of the initial state in a semilinear time-fractional subdiffusion equation from noisy random space-time point observations. A rescaled Gaussian prior based on a Whittle--Matérn process is assigned to the unknown initial condition. We prove that the solution has Ḣ3-regularity when the nonlinearity satisfies a Lipschitz condition in the Sobolev norm. We then establish posterior contraction rates for the prediction error in the L2-norm and for the parameter in Sobolev norms. The rates are polynomial in the sample size, with exponent depending on the prior smoothness and the spatial dimension. Moreover, we prove a minimax lower bound by constructing a wavelet-packing set and controlling the Kullback--Leibler divergences.

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Cite

BibTeXRIS
BibTeX
@misc{lu2026posterior,
  title = {{Posterior Consistency for Recovering Initial States in Nonlinear Subdiffusion Equations}},
  author = {Haoyu Lu and Shaokang Zu and Junxiong Jia},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.26536},
  eprint = {2608.26536},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.26536},
}
RIS
TY  - UNPB
TI  - Posterior Consistency for Recovering Initial States in Nonlinear Subdiffusion Equations
AU  - Haoyu Lu
AU  - Shaokang Zu
AU  - Junxiong Jia
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2608.26536
UR  - https://arxiv.org/abs/2608.26536
AB  - We study the Bayesian recovery of the initial state in a semilinear time-fractional subdiffusion equation from noisy random space-time point observations. A rescaled Gaussian prior based on a Whittle--Matérn process is assigned to the unknown initial condition. We prove that the solution has Ḣ^3-regularity when the nonlinearity satisfies a Lipschitz condition in the Sobolev norm. We then establish posterior contraction rates for the prediction error in the L^2-norm and for the parameter in Sobolev norms. The rates are polynomial in the sample size, with exponent depending on the prior smoothness and the spatial dimension. Moreover, we prove a minimax lower bound by constructing a wavelet-packing set and controlling the Kullback--Leibler divergences.
ER  -