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The coefficients in a general second order linear stochastic partialdifferential equation (SPDE) are estimated from multiple spatially localisedmeasurements. Assuming that the spatial resolution tends to zero and the numberof measurements is non-decreasing, the rate of convergence for each coefficientdepends on the order of the parametrised differential operator and is fasterfor higher order coefficients. Based on an explicit analysis of the reproducingkernel Hilbert space of a general stochastic evolution equation, a Gaussianlower bound scheme is introduced. As a result, minimax optimality of the ratesas well as sufficient and necessary conditions for consistent estimation areestablished.

Optimal parameter estimation for linear SPDEs from multiple measurements