On the probabilistic representation of the resolvent of the two-dimensional Laplacian

In this paper we consider a family of random linear operators that arises in the construction of a probabilistic representation of the resolvent of the two-dimensional Laplacian. It is shown that with probability $1$ 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.