Nonparametric velocity estimation in stochastic convection-diffusion equations from multiple local measurements

TL;DR

This paper develops a nonparametric method for estimating the velocity field in stochastic convection-diffusion equations using multiple local measurements, achieving minimax-optimal convergence rates.

Contribution

It introduces a weighted augmented MLE approach for pointwise velocity estimation from local measurements in SPDEs, establishing its convergence and optimality.

Findings

Achieves minimax-optimal convergence rates for velocity estimation.

Validates the method's optimality via lower bound analysis.

Demonstrates effectiveness with multiple local measurements.

Abstract

We investigate pointwise estimation of the function-valued velocity field of a second-order linear SPDE. Based on multiple spatially localised measurements, we construct a weighted augmented MLE and study its convergence properties as the spatial resolution of the observations tends to zero and the number of measurements increases. By imposing H\"older smoothness conditions, we recover the pointwise convergence rate known to be minimax-optimal in the linear regression framework. The optimality of the rate in the current setting is verified by adapting the lower bound ansatz based on the RKHS of local measurements to the nonparametric situation.