Inverting weak random operators

TL;DR

This paper explicitly finds the inverse and Green kernel of weak random operators, enabling spectral analysis and establishing strong operators using bilinear forms, Sturm-Liouville theory, and stochastic calculus.

Contribution

It provides explicit inverses and Green kernels for weak random operators, advancing their spectral analysis and connection to strong operators.

Findings

Explicit inverse and Green kernel formulas derived.

Spectral properties of the operators analyzed.

Existence of associated strong operators established.

Abstract

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.