  In Rajeev (2013), 'Translation invariant diffusion in the space of tempered
distributions', it was shown that there is an one to one correspondence between
solutions of a class of finite dimensional SDEs and solutions of a class of
SPDEs in $\mathcal{S}'$, the space of tempered distributions, driven by the
same Brownian motion. There the coefficients $\bar{\sigma}, \bar{b}$ of the
finite dimensional SDEs were related to the coefficients of the SPDEs in
$\mathcal{S}'$ in a special way, viz. through convolution with the initial
value $y$ of the SPDEs.
  In this paper, we consider the situation where the solutions of the finite
dimensional SDEs are stationary and ask whether the corresponding solutions of
the equations in $\mathcal{S}'$ are also stationary. We provide an affirmative
answer, when the initial random variable takes value in a certain set
$\mathcal{C}$, which ensures that the coefficients of the finite dimensional
SDEs are related to the coefficients of the SPDEs in the above `special'
manner.
