Only problem is that your nonlinear term is in fact not of gradient form from an infinite-dimensional point of view. Still, the reference I mentioned may be useful.

A good reference for this, if you haven't already found it, may be Da Prato & Zabczyk, 2nd Order PDEs in Hilbert Spaces, Chapter 12 (Gradient systems), although it considers the case where the state space is Hilbert $H$ and not Banach as in your case. If I understand correctly, the Markov generator is then defined on a space $\mathcal E_A(H)$ of functions of the form $e^{i \langle h, x \rangle}$ which may give a suggestion for your case. Also the book lists many references. A selection of this material can also be found in Da Prato, An introduction to infinite-dimensional analysis (2006).

@Nate Eldredge: in the introduction (page xxviii) of the reference CodeGolf mentions, it is stated that for a given family of marginals $(\mu_t)$, under some conditions uniqueness of the associated martingale can be shown, but not in general. In fact the authors state that they provide numerous examples within the book where uniqueness does not hold (but I did not look so close as to find these examples). A reference for the uniqueness result is G. Lowther, "Fitting martingales to given marginales", 2008.

We wish to show tightness of $\mathcal M_{\Omega}(\mu)$. By Doob's submartingale inequality, we can establish uniform boundedness of the trajectories in probability using only the marginal at time $t = 1$. Then if trajectories are Hölder continuous almost surely, i.e. $|X(t) - X(s)| \leq M |t-s|^{\alpha}$, for some fixed constants $M$ and $0 < \alpha \leq 1$, we have by the Arzelà-Ascoli theorem that uniformly bounded sets of $\alpha$-Hölder continuous functions are relatively compact. The question then is whether we can show this kind of regularity (uniformly in probability is sufficient).

@Misha: do I understand correctly that you mean simultaneous diagonalization by congruence, as described e.g. in Horn, Johnson - Matrix Analysis, Section 4.5 (in particular Theorem 4.5.15)?

What is meant by $|\mathbf P|$? The cardinality (this I assumed in my answer below), or the 'size' of a 'supremal' element in $\mathbf P$ (this would require a precise definition)?

I think there are some issues with your problem statement. You cannot sample points with uniform distribution in $\mathbb R^d$. Also, what do you mean by "achieving a density"?

Perhaps you are interested in the notion of "mutual information" (en.wikipedia.org/wiki/Mutual_information), which has a general definition as the relative entropy of the product distribution of the marginals, with respect to the joint distribution. It seems to be in the spirit of what you are asking.

I corrected my mistake.

I don't know what happened but I seems I voted this down by mistake (instead of up), and now I cannot repair this as long as the post is not edited. So think (-(-1) + 1) = +2! Or, if you wish to make the effort, make a slight edit so I can correct my mistake (I don't want to interfere with your post myself.) My apologies!!