Existence of the inverse of a linear stochastic operator
This paper shows that the inverse \(L^{-1}\) can be computed where L is an nth-order linear stochastic differential operator in \(Ly=x\). Somehow the assumption made in the proof that initial conditions were zero disappeared in the publication process. With this necessary statement, the result is correct. It is best to write equation (1) of the paper as \(y=\sum^{\infty}_{n=0}(-1)^ n(L^{-1}R)^ ny_ 0\), where \(y_ 0=\phi +L^{-1}x\) with \(\phi\) representing the homogeneous solution. This yields a correct solution for \(Ly=x\) but then we cannot say \(L^{- 1}\) is the inverse. We can generalize the computation to the two-, three- , or four-dimensional case, again getting an algorithm for solution (even for a nonlinear case if the first author's \(A_ n\) polynomials are used for nonlinear terms), however not an inverse unless all initial/boundary conditions vanish.
scientific article; zbMATH DE number 3960700