# Nonparametric Inference for Noise Covariance Kernels in Parabolic SPDEs using Space-Time Infill-Asymptotics

## TL;DR

This paper introduces a new asymptotic theory for nonparametric estimation of noise covariance kernels in parabolic SPDEs, enabling reliable inference with various spatial sampling schemes.

## Contribution

It develops a novel asymptotic limit theory for nonparametric covariance estimation in SPDEs using space-time infill asymptotics, independent of the differential operator.

## Key findings

- Consistent estimation of noise covariance kernels under mild regularity conditions.
- Construction of omnibus goodness-of-fit tests for the noise covariance.
- Framework accommodates diverse spatial sampling schemes and coarser spatial resolutions.

## Abstract

We develop an asymptotic limit theory for nonparametric estimation of the noise covariance kernel in linear parabolic stochastic partial differential equations (SPDEs) with additive colored noise, using space-time infill asymptotics. The method employs discretized infinite-dimensional realized covariations and requires only mild regularity assumptions on the kernel to ensure consistent estimation and asymptotic normality of the estimator. On this basis, we construct omnibus goodness-of-fit tests for the noise covariance that are independent of the SPDE's differential operator. Our framework accommodates a variety of spatial sampling schemes and allows for reliable inference even when spatial resolution is coarser than temporal resolution.