Nonparametric Diffusivity Estimation for the Stochastic Heat Equation from Noisy Observations

We estimate nonparametrically the spatially varying diffusivity of a stochastic heat equation from observations perturbed by additional noise.

To that end, we employ a two-step localization procedure, more precisely, we combine local state estimates into a locally linear regression approach.

Our analysis relies on quantitative Trotter--Kato type approximation results for the heat semigroup that are of independent interest.

The presence of observational noise leads to non-standard scaling behaviour of the model.

Numerical simulations illustrate the results.