  In this paper, we study the dynamics of the adjoint of a weighted composition
operator and we give necessary and sufficient conditions for this adjoint
operator to be topologically hyper-transitive on the space of Radon measures on
a locally compact Hausdorff space. Moreover, we provide sufficient conditions
for this operator to be chaotic and we give concrete examples. Next, we
consider the real Banach space of signed Radon measures and we give in this
context the sufficient conditions for the convergence of Markov chains induced
by the adjoint of an integral operator. Also, we illustrate this result with a
concrete example. In addition, we obtain some structural results regarding the
space of Radon measures. We characterize a class of cones whose complement is
spaceable in the space of Radon measures.
