Estimation for the reaction term in semi-linear SPDEs under small diffusivity

TL;DR

This paper develops a statistical framework for estimating reaction terms in semi-linear SPDEs with small diffusivity, providing CLTs, confidence intervals, and extensions to non-parametric and discrete observations using advanced stochastic analysis tools.

Contribution

It introduces a novel estimation method for reaction terms in semi-linear SPDEs under small diffusivity, including CLTs, efficiency results, and extensions to non-parametric and discrete data.

Findings

Central limit theorem for estimation error

Statistical efficiency via local asymptotic normality

Extension to non-parametric and discrete observations

Abstract

We consider the estimation of a non-linear reaction term in the stochastic heat or more generally in a semi-linear stochastic partial differential equation (SPDE). Consistent inference is achieved by studying a small diffusivity level, which is realistic in applications. Our main result is a central limit theorem for the estimation error of a parametric estimator, from which confidence intervals can be constructed. Statistical efficiency is demonstrated by establishing local asymptotic normality. The estimation method is extended to local observations in time and space, which allows for non-parametric estimation of a reaction intensity varying in time and space. Furthermore, discrete observations in time and space can be handled. The statistical analysis requires advanced tools from stochastic analysis like Malliavin calculus for SPDEs, the infinite-dimensional Gaussian Poincar\'e inequality and regularity results for SPDEs in $L^p$-interpolation spaces.