# Nonparametric estimation for linear SPDEs from local measurements

## TL;DR

This paper develops rate-optimal, robust nonparametric estimators for the leading coefficient of linear SPDEs using localized continuous observations, with theoretical analysis and numerical validation.

## Contribution

It introduces new nonparametric estimation methods for linear SPDEs from local measurements, achieving parametric rates and robustness to perturbations.

## Key findings

- Establishes asymptotic properties of estimators in fixed time and shrinking spatial resolution regimes.
- Provides scaling limits of the PDE and SPDE on expanding domains.
- Demonstrates estimator robustness to lower order operator perturbations.

## Abstract

The coefficient function of the leading differential operator is estimated from observations of a linear stochastic partial differential equation (SPDE). The estimation is based on continuous time observations which are localised in space. For the asymptotic regime with fixed time horizon and with the spatial resolution of the observations tending to zero, we provide rate-optimal estimators and establish scaling limits of the deterministic PDE and of the SPDE on growing domains. The estimators are robust to lower order perturbations of the underlying differential operator and achieve the parametric rate even in the nonparametric setup with a spatially varying coefficient. A numerical example illustrates the main results.