  Some linear integro-differential operators have old and classical
representations as the Dirichlet-to-Neumann operators for linear elliptic
equations, such as the 1/2-Laplacian or the generator of the boundary process
of a reflected diffusion. In this work, we make some extensions of this theory
to the case of a \emph{nonlinear} Dirichlet-to-Neumann mapping that is
constructed using a solution to a \emph{fully nonlinear} elliptic equation in a
given domain, mapping Dirichlet data to its normal derivative of the resulting
solution. Here we begin the process of giving detailed information about the
L\'evy measures that will result from the integro-differential representation
of the Dirichlet-to-Neumann mapping. We provide new results about both linear
and nonlinear Dirichlet-to-Neumann mappings. Information about the L\'evy
measures is important if one hopes to use recent advancements of the
integro-differential theory to study problems involving Dirichlet-to-Neumann
mappings.
