According to this reference page
StationaryDistribution[proc] represents the stationary distribution of the process proc, when it exists.
When I define the OrnsteinUhlenbeckProcess by the equivalent ItoProcess, SliceDistribution and StationaryDistribution are not found analytically any more. See for example the following:
In[25]:= SliceDistribution[OrnsteinUhlenbeckProcess[0, 1, 1], t]
Out[25]= NormalDistribution[0, 1/Sqrt[2]]
In[27]:= SliceDistribution[
ItoProcess[\[DifferentialD]x[
t] == -x[t] \[DifferentialD]t + \[DifferentialD]w[t],
x[t], {x, 1}, t, w \[Distributed] WienerProcess[]], t]
Out[27]= ItoProcess[{{-x[t]}, {{1}}, x[t]}, {{x}, {1}}, {t, 0}][t]