In these notes, D. Stroock gives a quick and efficient construction of the Markov transition functions of a certain diffusion. The idea of his construction (on page 4) is to 'freeze' the co-efficients of the Kolmogorov forward equation during "inter-dyadic" time-intervals $[m2^{-n},(m+1)2^{-n})$ whereby he obtains an approximate transition functions $P_n(x_1|x_2,t)$ simply by an extended convolution of Gaussians. Towards the bottom third of p.4 he says:
Using elementary facts about weak convergence of probability measures, one can show that there is a continuous map $(t, x)|\mapsto P(x_0|x,t)$ such that $P_n(x_0|.,t) → P(x_0|.,t)$ uniformly on compacts.
Concerning these "elementary facts", I am able to establish that the family $\{P_n(x_0|.,t)\}_n$ will be tight and therefore will have accumulation points in the Lévy-Prokhorov metric (Prokhorov's theorem). But what does it take to establish that there's only one accumulation point? How do you show that?
UPDATE: I think I got it already.
Let $(n_k)_k$ be a subsequence so that $P_{n_k}(x_0|C,t)\to P(x_0|C,t)$ for every compact $C$. ($P(x_0|C,t)$ is shorthand for $\int dy\,P(x_0|y,t)\chi_{y\in C}$. The claimed subsequence exists by Prokhorov's theorem)
(Local) bounds on coefficient functions and compactness of $C$ imply that $P_{n_k}(x_0|C,t)$ cannot vary too much during the inter-dyadic time-interval $[m2^{-n_k},(m+1)2^{-n_k})$ and this implies that for all $n\in \{n_k,...,n_{k+1}\}$: $P_{n_k}(x_0|C,t)\approx P_{n}(x_0|C,t)\approx P_{n_{k+1}}(x_0|C,t)$, i.e. the whole sequence $(P_{n}(x_0|C,t))_n$ converges.
Finishing touch: If we have fixed $P(x_0|C,t)$ on every compact $C$, a type of $\pi-\lambda$ theorem/monotone class theorem should uniquely fix $P(x_0|B,t)$ on all Borel sets $B$.