  The coefficients in a second order parabolic linear stochastic partial
differential equation (SPDE) are estimated from multiple spatially localised
measurements. Assuming that the spatial resolution tends to zero and the number
of measurements is non-decreasing, the rate of convergence for each coefficient
depends on its differential order and is faster for higher order coefficients.
Based on an explicit analysis of the reproducing kernel Hilbert space of a
general stochastic evolution equation, a Gaussian lower bound scheme is
introduced. As a result, minimax optimality of the rates as well as sufficient
and necessary conditions for consistent estimation are established.
