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For the past few days I have been studying Advanced statistical mechanics. I am studying a Wiener process in general. Such a process is a non-stationaty time-independent Gaussian process. The probability density at a time t is given by $$\rho_t(x) = \frac{1}{\sqrt{4\pi Dt}}e^{\frac{-x^2}{4Dt}}.$$ I am now wondering how you get the transition probability density from this which is, in this case, given by $$\rho_t(x,y) = \frac{1}{\sqrt{4\pi Dt}}e^{\frac{-(x-y)^2}{4Dt}}.$$ A quick look makes me think you should just replace $x$ by $x-y$, however my question is more general in the sense that what if I have some probability distribution how do I compute the transition probability distribution from this?

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There are many possibilities. If I understand correctly, you are asking given a distribution, how to get a Markov chain (like in your example, or a stochastic process more generally) which leaves it invariant. There is no canonical way to do so.

There exists many standard recipes depending on your interest. The most famous one is the Metropolis-Hastings algorithm, which is useful for sampling the original distribution.

Hope this helps.

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