  We analyze two weak random operators, initially motivated from processes in
random environment. Intuitively speaking these operators are ill-defined, but
using bilinear forms one can deal with them in a rigorous way. This point of
view can be found for instance in the work Skorohod \cite{Skorohod}, and it
remarkably helps to carry out specific calculations. In this paper, we find
explicitly the inverse of such weak operators, by provinding the forms of the
so-called Green kernel. We show how this approach helps to analyze the spectra
of the operators. In addition, we provide the existence of strong operators
associated to our bilinear forms. Important tools that we use are the
Sturm-Liouville theory and the stochastic calculus.
