  In this paper we calculate the vacuum expectation values of the stress-energy
bitensor of a massive quantum scalar field with general coupling to
N-dimensional Euclidean spaces and hyperbolic spaces which are Euclidean
sections of the anti-de Sitter (AdS) spaces. These correlators, also known as
the noise kernel, act as sources in the Einstein-Langevin equations of
stochastic gravity [1,2] which govern the induced metric fluctuations beyond
the mean-field dynamics described by the semiclassical Einstein equations of
semiclassical gravity. Because these spaces are maximally symmetric the
eigenmodes have analytic expressions which facilitate the computation of the
zeta-function [3,4]. Upon taking the second functional variation of the
generalized zeta function introduced in [5] we obtain the correlators of the
stress tensor for these two classes of spacetimes. Both the short and the large
geodesic distance limits of the correlators are presented for dimensions up to
11. We mention current research problems in early universe cosmology, black
hole physics and gravity-fluid duality where these results can be usefully
applied to.
