Spectral asymptotics of some functionals arising in statistical inference for SPDEs

A parameter estimation problem is considered for a stochastic evolution equation on a compact smooth manifold. Specifically, we concentrate on asymptotic properties of spectral estimates, i.e. estimates based on finite number of spatial Fourier coefficients of the solution. Under certain non-degeneracy assumptions the estimate is proved to be consistent, asymptotically normal and asymptotically efficient as the dimension of the projections increases. Unlike previous works on the subject, no commutativity is assumed between the operators in the equation.