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On p.4 of 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 coefficients of the Kolmogorov forward equation during "inter-dyadic" time-intervals $[m2^{-n},(m+1)2^{-n})$ whereby he obtains an approximate transition functions $P_n(x_2|x_1,t)$ 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 $(t, x)|\mapsto P(.|x_0,t)$ such that $P_n(.|x_0,t) → P(.|x_0,t)$ uniformly on compacts.

I think I can follow him in that rush of thought insofar that I believe the family $\{P_n(.|x_0,t)\}_n$ is easily shown to be tight. Prokhorov's theorem can subsequently get some things done: there exists a weakly converging subsequence of the measures $P_n(x|x_0,t)dx$. Subsequently, I see a way to show that in fact the sequence must converge in its entirety. I however don't see how he finds that the limit has a continuous density $P(.|x_0,t)$ and how $P_n(.|x_0,t) \to P(.|x_0,t)$ uniformly on compacts. I know that it could be accomplished if one uses Arzelà-Ascoli, which requires as a pre-condition the equicontinuity and boundedness of the functions $\{P_n(.|x_0,t)\}_n$, which I don't know how to prove and which don't seem to fit well under the nomer "elementary facts about weak convergence" so that I think that I might be on the wrong track.

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  • $\begingroup$ I don't think that he ever claims that the $P(t,x)$ have a continuous density, but that the continuity he's referring to is the continuity $(t,x) \mapsto P(t,x)$ from $\mathbb{R}^{N+1}$ into the space of probability measures (with the topology of weak convergence). $\endgroup$
    – Martin Hairer
    Commented Mar 2, 2020 at 21:48
  • $\begingroup$ I see, but why is the continuity of that dependence on $(t,x)$ and its uniform convergence on compact $x$-regions relevant in the immediate sequel of the discussion? I.e. he writes " Moreover, because, uniformly on compacts, $\langle L_y\phi, \Gamma(\{\tau\}_n,y) \rangle \to L\phi$". I don't see why you have to bother about the $x$-dependence to establish that fact (rather the increasing concentration of $\Gamma(\{\tau\}_n,y)$ seems what matters). $\endgroup$
    – 5th decile
    Commented Mar 2, 2020 at 22:10
  • $\begingroup$ I may again misunderstand his purposes (or just be mistaken). But I see no other occurrence in the subsequent discussion where facts about the dependence on $(t,x)$ seem to matter. $\endgroup$
    – 5th decile
    Commented Mar 2, 2020 at 22:12

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