Abstract

In this talk we consider the estimation of the diffusivity in a linear stochastic heat equation with additive noise. The measurements are spatial averages of the process at different locations. Each measurement corresponds to the spatial average of the process relative to a known point spread kernel function with small support observed continuously in time over a fixed time interval. The process is therefore only partially observed (in space) and the resulting vector-valued observation process is non-Markovian. We nevertheless show that it is possible to verify the LAN (local asymptotic normality) property of the likelihood process, as the support radius shrinks to zero and the number of measurements increases. The proof is inspired by the classical Kalman-Bucy filter.