Locality and Domination of Semigroups

We characterize all semigroups $(T(t){)}_{t\ge 0}$ on $L^2(\mathrm{\Omega })$ sandwiched between Dirichlet and Neumann ones, i.e.: \begin{equation*}\label{eq:san} e^{t\Delta_D}\leq T(t)\leq e^{t\Delta_N}\quad,$\text{for all }$t\geq0 \end{equation*} in the positive operators sense. The proof uses the well-known Beurling-Deny and Lejan formula to drop the locality assumption made usually on the form associated with $(T(t){)}_{t\ge 0}$.