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I was looking at the Theorem of the seminal paper by Jordan "The variational formulation of the Fokker-Planck equation" and in their "Theorem" in section 5 I was a bit confused by their setup. As I understood it, they are looking to prove weak convergence in $L^1(\mathbb{R}^n)$ of their interpolation $\rho_h dx$ to $\rho$ which satisfies the Fokker plank. According to this post Weak convergence in $L^p$ space., this amounts to showing that for all $\varphi \in L^\infty(\mathbb{R}^n)$, $$\lim_{n \to \infty}\int_{\mathbb{R}^n} \rho_h \varphi dx\to \int_{\mathbb{R}^n} \rho \varphi dx $$

However, in this paper they started with some vector field $\xi$ and the resulting flow function $\Phi$. I'm not sure why they have this and how this relates to the test function $\varphi$.

On a marginally related note towards the end of the proof (top of page 21) they write "Owing to the estimates (42) and (43), we may conclude that there exists a measurable $\rho(t, x)$ such that, after extraction of a subsequence, $$\rho_h \to \rho \quad \text{ weakly in } L^1\left((0, T) \times \mathbb{R}^n\right) \text{for all } T<\infty."$$ How do they know this is true? I know that this is a property of compact sets (closed and bounded) so I thought their estimates which were showing something is bounded might played a role but I'm not sure.

In summary, while some of the details make sense, I'm getting lost in them and struggling to unterstand the bigger picture of the proof

Math_Day
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  • Not entirely sure, but I took a quick look. The main estimates all come together just after equation $(49)$, which involves 3 main terms, the fact that each of these are small come from the smoothness of $\rho$, $(47)$, and $(49)$. The proof starts with estimates with $\xi$ because they are setting up to prove the convergence to the solution of the FP equation. I'm not entirely sure on your second comment but the proof likely follows from Prokhorov's theorem, then diagonalization to get convergence on all rationals, then some time continuity argument. I'll take a longer look in a moment. – person Mar 20 '25 at 00:43
  • Wait actually I'm losing it. No need for such a complicated argument to justify their weak convergence claim, all you need is the Dunford-Pettis theorem for weak convergence, the set $\rho_h$ is uniformly integrable by $(42)$. – person Mar 20 '25 at 01:17

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