Let $\mu\_t, t \geq 0,$ be a family of probability measures on the real line. One can assume whatever one wishes about them, although typically they will be continuous in some topology (usually at least the topology of weak convergence of measures), and they will be absolutely continuous with respect to Lebesgue measure. The basic question is as follows:


Is there a Markov process $X\_t$ such that its marginal distribution at each time is $\mu\_t$?


An obvious example is when $$d \mu\_t = \frac{e^{-x^2/2t}}{\sqrt{2 \pi t}} dx$$ and $\mu\_0 = \delta\_0$, in which case we know that Brownian motion is such a Markov process. I am curious to know if there is any general theory along these lines.


Edit
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As per Byron's comment below, I would like the Markov process to be continuous. Ideally I would like to have an SDE description of the process. 


The SDE description actually suggests one possible answer: simply compute and play with the time and space derivatives of the density function to see if they satisfy some sort of parabolic equation (like the heat equation), use this to get the adjoint of the generator, and then compute the generator itself. This is a very plausible option, but I was hoping that there might be something more systematic.



Markov Processes with Given Marginals
The following result was proved in 


Kellerer, H.G. (1972) Markov-Komposition und eine Anwendung auf Martingale. Math. Ann., 198,
99–122.


Let $p(y, t)$ be a family of marginal densities, with ﬁnite ﬁrst moment, such that
for $s , t$ the density at time $t$ dominates the density at time $s$ in the convex order. Then there exists a Markov process $X(t)$ with these marginal densities under which $X(t)$ is a submartingale. Furthermore, if the means are independent of $t$ then $X(t)$ is a martingale.


Constructive versions have then been studied by Madan and Yor in


<http://projecteuclid.org/download/pdf_1/euclid.bj/1078681382>


The SDE description you mention is actually discussed in Section 2.3 of the paper.


