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Non-parametric estimation for stochastic reaction-diffusion equations with spatial ergodicity

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  • Gaudlitz, Sascha

Abstract

The non-parametric estimation of a non-linear reaction function in a reaction-diffusion stochastic partial differential equation (SPDE) is discussed. The estimation error can be bounded in terms of the diffusivity and the noise level, which obey a realistic coupling and tend to zero. The estimator achieves the minimax-optimal convergence rate and exhibits asymptotic normality due to the spatial ergodicity of the SPDE. Analysing the estimation error requires the control of spatial averages of non-linear transformations of the SPDE, and combines the Clark-Ocone formula from Malliavin calculus with the Markovianity of the SPDE. In contrast to previous results, the obtained variance bound is uniform in the Lipschitz-constant of the transformation.

Suggested Citation

  • Gaudlitz, Sascha, 2026. "Non-parametric estimation for stochastic reaction-diffusion equations with spatial ergodicity," Stochastic Processes and their Applications, Elsevier, vol. 201(C).
    • Handle: RePEc:eee:spapps:v:201:y:2026:i:c:s0304414926001845
    DOI: 10.1016/j.spa.2026.105052
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