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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]
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1 Answer 1

up vote 1 down vote accepted

Even though the result is not displayed the way one would expect, the distribution is there :

PDF[SliceDistribution[
     ItoProcess[\[DifferentialD]x[t] == -x[t] \[DifferentialD]t + 
                 \[DifferentialD]w[t], x[t], {x, 1}, t, 
                w \[Distributed] WienerProcess[]], t], z]

$$\frac{\exp \left(-\frac{e^t \left(z-e^{-t}\right) \left(e^t z-1\right)}{e^{2 t}-1}\right)}{\sqrt{\pi } \sqrt{e^{-2 t} \left(e^{2 t}-1\right)}}$$

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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. –  Tamas Mar 16 at 9:55
    
I found Limit[PDF[SliceDistribution[proc, t], x], t -> Infinity] to work, but I still have the question about StationaryDistribution. –  Tamas Mar 16 at 10:07

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