I am currently trying to understand the big picture/connections of Martingale Problem, Fokker-Planck-equations (although, until now, I have mostly kept these out of my considerations), SDEs and Markov processes. Maybe this long post is valuable for some people trying to get into this area and the connections, as I will try to point out as many connections and relations as possible. But I have some questions myself. Ok, here we go. The way I think about these items (on a large scale) is (roughly) as follows: (I only consider continuous Markov processes with values in $\mathbb{R}^d$)
The Markov process is really at the center of the attention - it is the ultimate object of desire, whereas the other ones are rather auxiliary tools for the understanding of it (I am very aware that these fields - in particular SDEs and, emerging from it, SPDEs have a large and rich theory of its own, which I am familiar with to a decent degree - but here I am really putting the Markov process in the heart of everything).
Assume we are in the lucky situation that we know the transition function/semigroup of the Markov Process or the family of its laws, usually called $(\mathbb{P}_x)_{x \in \mathbb{R}^d}$ AND we also know the generator. Then we have everything we could ask for and the corresponding Martingale problem or the corresponding SDE are not needed. Is this essentially correct?
Now of course one hardly ever has all these information at hand. A much more common situation is/seems to be (I have not witnessed this myself, but read it several times) that one only knows the generator, which is of the form $$L(x) = \frac{1}{2}a_{ij}(x)\partial_i\partial_j+b_i(x)\partial_i.$$Further, one might not even know whether the given operator generates a Markovian semigroup, i.e. a Markov process. Intuively, $L$ describes the drift (via $b$) and the variance/fluctuation (via $a$) of the particle governed by the Markov law under consideration when it sits at position $x$ (for simplicity, let's stick to the time-homogeneous case).
Well, knowing only the generator may not be sufficient for a thorough understanding of the processes behaviour - hence one turns to the corresponding SDE and/or the Martingale problem (as I said, I omit the FP-eq. for now) for more information.
$\textbf{SDE}:$ The corresponding stochastic differential equation reads $dX_t = b(X_t)dt+\sigma(X_t)dB_t$, where (if existing!) $\sigma\sigma^T = a$ is a (possibly non-unique) square root of $a$ and $B$ is a Brownian motion. It is well-known that if the coefficients of the SDE are suffiecently good to admit a unique strong solution, then this solution is a Markov process with $L$ as its generator. Then one may analyze this solution by means of SDE-/stochastic analysis techniques to obtain interesting properties.
However, the drawback of this approach lies in (at least) two points: First, obviously our coefficients might not be good enough, i.e. they might not be Lipschitz or similar. Then we don't have uniqueness and do not obtain the Markov property (right?). Secondly, the diffusion matrix $a$ may not have have square root $\sigma$ with such nice properties, even if $a$ itself enjoys good properties (I know there are some conditions under which one can transfer properties of $a$ to its square root. But it is not always the case). Hence it is natural to look for another approach. This is why Stroock\Varadhan introduced the
$\textbf{Martingale Problem}:$ First of all it is quite natural to study it, because by Ito-formula every solution to an SDE (in the sense of its law) as above solves the corresponding martingale problem ($\textit{corresponding}$ means: The coefficients of the martingale problem are $b$ and a = $\sigma\sigma^T$). Now appparently I can write down the martingale problem in a meaningful way without any assumptions on the coefficients but measurability - an advantage compared to the SDE-apporach! Also, a priori I do not have to care about a square root of $a$ with good properties. Nice!
$\textbf{BUT}$, and now comes my actual main question, is the martingale problem-apporach in the end really more general? I KNOW that the two approaches are equivalent if I know that my $a$ has a sufficiently nice square root. So for justification, there should be a real benefit for the martingale problem-apporach in the case where $a$ or its square root(s) are not so nice. Hence let us assume I have coefficients $a$ and $b$, which are not good enough for the SDE-approach. What then? The martingale problem-approach is only beneficial, if I know that it is well-posed, for then the existence of a Markov process (the $\mathbb{P}_x$ are the unique solutions to the martingale problem in this case) with the given generator follows (else you may still be able to make a Markovian selection, c.f. Chapter 12 S.\V., but I dont think you can get many information out of this selection, can you?)
So, to wrap it up: Is, and if yes in what precise sense, the martingale problem apporach in the end really strictly better than the SDE-approach? I feel I really need to get this straight to arrange a better structure/hierarchy in my head.
I am grateful for any reasonable comment on this - an answer to any of my side question or just further discussion!