  We introduce the transportation-annihilation distance $W_p^\sharp$ between
subprobabilities and derive contraction estimates with respect to this distance
for the heat flow with homogeneous Dirichlet boundary conditions on an open set
in a metric measure space.
  We also deduce the Bochner inequality for the Dirichlet Laplacian as well as
gradient estimates for the associated Dirichlet heat flow.
  For the Dirichlet heat flow, moreover, we establish a gradient flow
interpretation within a suitable space of charged probabilities.
  In order to prove this, we will work with the \emph{doubling} of the open
set, the space obtained by gluing together two copies of it along the boundary.
