On the Properties of a Class of Random Operators

We consider random operators arising when one constructs a probabilistic representation of the resolvent of an operator \(-\frac{1}{2}\frac{d}{dx}\left({b}^{2}\left(x\right)\frac{d}{dx}\right)+V\left(x\right)\) $-\frac{1}{2}\frac{d}{\mathit{dx}}\left(b^2,\left(x\right),\frac{d}{\mathit{dx}}\right)+V\left(x\right)$ . We prove that with probability one these operators are linear integral operators and study properties of their kernels.