Estimation problems for coefficients of stochastic partial differential equations. Part I

This paper considers the problem of estimating functional parameters $a_k(t,x)$, $f(t,x)$ by observing a solution $u_{\varepsilon}(t,x)$ of a stochastic partial differential equation
$$ du_{\varepsilon}(t)=\sum_{|k|\le2p}a_kD_x^ku_{\varepsilon}+f\,dt+\varepsilon\,dw(t), $$
where $w(t)$ is a Wiener process. The asymptotic statement of the problem is considered when the noise level $\varepsilon\to0$. In the first part of the work we determine what is considered the statistics of the problem and investigate the problem of estimating $f$.