{"ID":2896595,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.06857","arxiv_id":"2507.06857","title":"Nonparametric Bayesian Inference for Stochastic Reaction-Diffusion Equations","abstract":"We consider the Bayesian nonparametric estimation of a nonlinear reaction function in a reaction-diffusion stochastic partial differential equation (SPDE). The likelihood is well-defined and tractable by the infinite-dimensional Girsanov theorem, and the posterior distribution is analysed in the growing domain asymptotic. Based on a Gaussian wavelet prior, the contraction of the posterior distribution around the truth at the minimax optimal rate is proved. The analysis of the posterior distribution is complemented by a semiparametric Bernstein--von Mises theorem. The proofs rely on the sub-Gaussian concentration of spatio-temporal averages of transformations of the SPDE, which is derived by combining the Clark-Ocone formula with bounds for the derivatives of the (marginal) densities of the SPDE.","short_abstract":"We consider the Bayesian nonparametric estimation of a nonlinear reaction function in a reaction-diffusion stochastic partial differential equation (SPDE). The likelihood is well-defined and tractable by the infinite-dimensional Girsanov theorem, and the posterior distribution is analysed in the growing domain asymptot...","url_abs":"https://arxiv.org/abs/2507.06857","url_pdf":"https://arxiv.org/pdf/2507.06857v1","authors":"[\"Randolf Altmeyer\",\"Sascha Gaudlitz\"]","published":"2025-07-09T13:58:51Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"math.PR\"]","methods":"[\"Diffusion Model\"]","has_code":false}