  We perform the point-splitting regularization on the vacuum stress tensor of
a coupling scalar field in de Sitter space under the guidance from the
adiabatically regularized Green's function. For the massive scalar field with
the minimal coupling $\xi=0$, the 2nd order point-splitting regularization
yields a finite vacuum stress tensor with a positive, constant energy density,
which can be identified as the cosmological constant that drives de Sitter
inflation. For the coupling $\xi\ne 0$, we find that, even if the regularized
Green's function is continuous, UV and IR convergent, the point-splitting
regularization does not automatically lead to an appropriate stress tensor. The
coupling $\xi R$ causes log divergent terms, as well as higher-order finite
terms which depend upon the path of the coincidence limit. After removing these
unwanted terms by extra treatments, the 2nd-order regularization for small
couplings $\xi \in(0,\frac{1}{7.04})$, and respectively the 0th-order
regularization for the conformal coupling $\xi=\frac16$, yield a finite,
constant vacuum stress tensor, in analogy to the case $\xi=0$. For the massless
field with $\xi=0$ or $\xi=\frac16$, the point-splitting regularization yields
a vanishing vacuum stress tensor, and there is no conformal trace anomaly for
$\xi=\frac16$. If the 4th-order regularization were taken, the regularized
energy density for general $\xi$ would be negative, which is inconsistent with
the de Sitter inflation, and the regularized Green's function would be singular
at the zero mass, which is unphysical. In all these cases, the stress tensor
from the point-splitting regularization is equal to that from the adiabatic
one.
