Parameter estimation for semilinear SPDEs from local measurements

Abstract

This work contributes to the limited literature on estimating the diffusivity or drift coefficient of nonlinear SPDEs driven by additive noise. Assuming that the solution is measured locally in space and over a finite time interval, we show that the augmented maximum likelihood estimator introduced in (Ann. Appl. Probab. 31 (2021) 1–38) for linear SPDEs remains rate-optimal when applied to a large class of semilinear SPDEs. The obtained abstract results are applied to several important classes of SPDEs, including stochastic reaction-diffusion equations. Moreover, we also study the stochastic Burgers equation, as an example with first order nonlinearity, which is a borderline case of the general results. The optimal statistical results are obtained through a precise control of the spatial regularity of the solution and by using higher order fractional Lp-Sobolev type spaces. We conclude with numerical examples that validate the theoretical results.