  In this note we contribute two results to the theory of the $2D$ Euler
equations in vorticity form on the full plane. First, we establish a
generalized Lagrangian representation of weak (in general measure-valued)
solutions, which includes and extends classical results on the Lagrangianity of
weak solutions. Second, we construct nonlinear Markov processes which are
uniquely determined by a selection of weak solutions from initial data in
$L^1\cap L^p$, $p \geq 2$, and related spaces such as the classical and
uniformly localized Yudovich space. It is well-known that for $p <\infty$ weak
solutions are in general not unique, which renders a suitable selection
nontrivial.
