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 doesn't seem to fit well under the nomer "easy facts about weak convergence" so that I think that I might be on the wrong track.

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.

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 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 doesn't seem to fit well under the nomer "easy facts about weak convergence" so that I think that I might be on the wrong track.

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 doesn't seem to fit well under the nomer "easy facts about weak convergence" so that I think that I might be on the wrong track.