Matching posterior contraction bounds for nonlinear subdiffusion initial-state recovery

Determine whether the upper and lower posterior contraction-rate bounds for recovering the initial state in the semilinear time-fractional subdiffusion equation match, thereby identifying the exact contraction rate as the sample size grows under the specified prior, estimator, and conditional stability estimate.

Background

The paper establishes an upper bound for posterior contraction in Sobolev norms of the unknown initial state and a separate minimax lower bound. The exponents in these bounds are not shown to coincide in general. The authors explain that whether the bounds match depends on the chosen prior, estimator, and conditional stability estimate for the forward map.

The remark places this unresolved issue in relation to prior results: matching bounds have been obtained in some parabolic inverse problems under Lipschitz stability, whereas non-matching bounds occur under Hölder or logarithmic stability. Determining whether matching bounds hold for the semilinear time-fractional subdiffusion initial-state recovery problem is therefore left as a concrete direction for future work.

References

Whether the upper and lower bounds on the contraction rate match is a challenging question, as it effectively asks for the exact rate as the sample size grows. This depends on the prior, the estimator, and the conditional stability estimate. Matching bounds were obtained in for the posterior mean under Lipschitz stability with a truncated Gaussian prior, while found that under Hölder stability the same prior and estimator do not yield matching bounds. observed non-matching bounds under logarithmic stability, whereas proved matching bounds for the MAP estimator under Lipschitz stability with a Gaussian prior. Whether matching bounds hold for the subdiffusion problem studied here will be an interesting topic for future work.

— Posterior consistency for subdiffusion inverse problems  (2608.26536 - Lu et al., 27 Aug 2026) in Remark following Theorem Minimax lower bound, Section 3 (Upper and lower bound of the posterior contraction rate)