# How to obtain SliceDistribution or StationaryDistribution for an ItoProcess when it is known to exist?

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]


PDF[SliceDistribution[

$$\frac{\exp \left(-\frac{e^t \left(z-e^{-t}\right) \left(e^t z-1\right)}{e^{2 t}-1}\right)}{\sqrt{\pi } \sqrt{e^{-2 t} \left(e^{2 t}-1\right)}}$$
• Thanks a lot for your answer. To do with this issue, I suppose, SliceDistribution[proc,Infinity] does not work, neither does StationaryDistribution[proc]. But is there really not a way to use StationaryDistribution only? I'm thinking that may be there are cases when SliceDistribution does not exists, but StationaryDistribution does. Mar 16 '14 at 9:55
• I found Limit[PDF[SliceDistribution[proc, t], x], t -> Infinity] to work, but I still have the question about StationaryDistribution. Mar 16 '14 at 10:07