Parameter estimation for stochastic evolution equations with non-commuting operators (Q2726270)

Parameter estimation is a particular case of the inverse problem when the solution of a certain equation is observed and conclusions must be made about the coefficients of the equation. In the deterministic setting, numerous examples of such problems in ecology, material sciences, biology, etc. are given in the book by \textit{H. T. Banks} and \textit{K. Kunisch} [``Estimation techniques for distributed parameter systems'' (1989; Zbl 0695.93020)]. The stochastic term is usually introduced in the equation to take into account those components of the model that cannot be described exactly. NEWLINENEWLINENEWLINEIn an abstract setting the parameter estimation problem is considered for an evolution equation NEWLINE\[NEWLINEdu(t)+(A_{0}+\theta A_{1})u(t)dt=\varepsilon dW(t),\;0< t\leq T;\quad u(0)=0,NEWLINE\]NEWLINE where \(\theta\) is an unknown parameter belonging to an open subset of the real line and \(W=W(t)\) is a random perturbation. If \(u\) is a random field, then a computable estimate of \(\theta\) must be based on finite-dimensional projections of \(u\) even if the whole trajectory is observed. The objective of this paper is to consider an estimate of \(\theta\) for the above-presented equation for \(u(t)\) without assuming anything about the eigenfunctions of the operators in the equation. The equation is considered on a compact smooth \(d\)-dimensional manifold so that there are no boundary conditions involved. The main assumption is that the operators \(A_{0}\) and \(A_{1}\) are of different orders and the operator \(A_{0}+\theta A_{1}\) is elliptic for all admissible values of \(\theta.\) Unlike previous works on the subject, no commutativity is assumed between the operators in the equation. The estimate is based on finite-dimensional projections of the solution. Under certain non-degeneracy assumptions the estimate is proved to be consistent and asymptotically normal as the dimension of the projections increases.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00022].