On the probabilistic representation of the resolvent of the two-dimensional Schrödinger operator

We consider a family of random linear operators that arises in the construction of a probabilistic representation of the resolvent of the two-dimensional Schrödinger operator. It is shown that with probability one the operators of this family are integral operators in $L_2(\mathbb{R}^2)$. The properties of the kernels of the corresponding operators are also investigated.