  Let $\Omega\subset{\mathbb R}^n$ be bounded with a smooth boundary $\Gamma$
and let $S$ be the symmetric operator in $L^2(\Omega)$ given by the minimal
realization of a second order elliptic differential operator. We give a
complete classification of the Markovian self-adjoint extensions of $S$ by
providing an explicit one-to-one correspondence between such extensions and the
class of Dirichlet forms in $L^2(\Gamma)$ which are additively decomposable by
the bilinear form of the Dirichlet-to-Neumann operator plus a Markovian form.
By such a result two further equivalent classifications are provided: the first
one is expressed in terms of an additive decomposition of the bilinear forms
associated to the extensions, the second one uses the additive decomposition of
the resolvents provided by Krein's formula. The Markovian part of the
decomposition allows to characterize the operator domain of the corresponding
extension in terms of Wentzell-type boundary conditions. Some properties of the
extensions, and of the corresponding Dirichlet forms, semigroups and heat
kernels, like locality, regularity, irreducibility, recurrence, transience,
ultracontractivity and Gaussian bounds are also discussed.
