  We study the relation between two sets of correlators in interacting quantum
field theory on de Sitter space. The first are correlators computed using in-in
perturbation theory in the expanding cosmological patch of de Sitter space
(also known as the conformal patch, or the Poincar\'e patch), and for which the
free propagators are taken to be those of the free Euclidean vacuum. The second
are correlators obtained by analytic continuation from Euclidean de Sitter;
i.e., they are correlators in the fully interacting Hartle-Hawking state. We
give an analytic argument that these correlators coincide for interacting
massive scalar fields with any $m^2 > 0$. We also verify this result via direct
calculation in simple examples. The correspondence holds diagram by diagram,
and at any finite value of an appropriate Pauli-Villars regulator mass M. Along
the way, we note interesting connections between various prescriptions for
perturbation theory in general static spacetimes with bifurcate Killing
horizons.
