Time invarient pdf but nonstationary process

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A solution to the general stochastic differential equation (SDE) can yield a time-independent probability density function (pdf) while maintaining a nonstationary stochastic process. It is possible to have a constant distribution function with a correlation function that depends on both time variables, indicating nonstationarity. An example provided is a Gaussian process with a mean of zero and standard deviation of one, where the correlation function is defined as f(s,t) = 1/(1+|s²-t²|). The discussion raises questions about how to compute this correlation and the apparent contradiction of having a time-independent pdf alongside a time-dependent correlation function. Understanding these concepts is crucial for exploring the dynamics of stochastic processes.
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I am wondering if there exist some solution to the general stochastic differential equation (SDE) such that I get a time independent pdf(x) while the stochastic process Xt is nonstationary.. I really need some help with that..
 
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I am not sure what you mean by a general stochastic diff. eq. However, it is possible to have a stochastic process with a constant distribution function, but where the correlation function is dependent on both values of the independent (time) variable, and not just the difference - therefore not stationary.
 
OK, forget about the SDE .. can u give me example of a stochastic process such that the pdf (dosen't depend on time == > dp/dt=0) but the correlation has a time variable (nonstationary)? this will helpso much..
 
Gaussian process (mean=0, s.d=1) with a correl. dep. on both variables. For example f(s,t)=1/(1+|s2-t2|).
 
what's f(s,t)..
can u please tell me how to compute this correlation? if the pdf has not time in it, how come time appears in the correlation function?
 
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