  We employ stochastic quantization for a self-interacting nonminimal massive
scalar field in curved spacetime. The covariant background field method and
local momentum space representation are used to obtain the Euclidean
correlation function and evaluate multi-loop quantum corrections through
simultaneous expansions in the curvature tensor and its covariant derivatives
and in the noise fields. The stochastic correlation function reproduces the
well-known result by Bunch and Parker and is used to construct the effective
potential in curved spacetime in an arbitrary dimension $D$ up to the first
order in curvature. Furthermore, we present a sample of numerical simulations
for $D=3$ in the first order in curvature.
