Sequential compactness of a sequence of curves of Borel probability measures

Question

$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \DeclareMathOperator*{\supp}{supp} \newcommand{\bR}{\mathbb{R}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bD}{\mathbb{D}} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\cA}{\mathcal{A}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cD}{\mathcal{D}} \newcommand{\cE}{\mathcal{E}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cG}{\mathcal{G}} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\sD}{\mathscr{D}} \newcommand{\sE}{\mathscr{E}} \newcommand{\sG}{\mathscr{G}} \newcommand{\sH}{\mathscr{H}} \newcommand{\sK}{\mathscr{K}} \newcommand{\sP}{\mathscr{P}} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\eps}{\varepsilon} \newcommand{\diff}{\mathop{}\!\mathrm{d}} \newcommand{\andd}{\quad \text{and} \quad} \newcommand{\qtext}{\quad\text} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $ Let $\sP_2 (\bR^d)$ be the space of Borel probability measure on $\bR^d$ with finite second moment.

• We denote by $M_2 (\nu) := \int_{\bR^d} |x|^2 \diff \nu (x)$ the second moment of $\nu \in \sP_2 (\bR^d)$. We denote by $W_2$ the $2$-Wasserstein metric on $\sP_2 (\bR^d)$.

• We denote by $\rightharpoonup$ the weak convergence in $\sP_2 (\bR^d)$, i.e., $\nu_n \rightharpoonup \nu$ in $\sP_2 (\bR^d)$ if and only if $\int_{\bR^d} f \diff \nu_n \to \int_{\bR^d} f \diff \nu$ for every $f \in C_b (\bR^d)$. Correspondingly, we denote by $\tau_{\mathrm w}$ the topology that $\rightharpoonup$ induces on $\sP_2 (\bR^d)$.

For each $n \in \bN^*$ and $t \in [0, 1]$, let $\mu^n_t \in \sP_2 (\bR^d)$. We fix $\alpha \in (0, 1)$ and assume that \begin{equation} \sup_{t \in [0, 1]} \sup_{n \in \bN^*} M_2 (\mu^n_t) + \sup_{n \in \bN^*} \sup_{\substack{s,t \in [0, 1] \\ s \neq t}} \frac{W_2 (\mu^n_t, \mu^n_s)}{|t-s|^\alpha} < \infty. \end{equation}

Then for each $t \in [0, 1]$, the sequence $(\mu^n_t)_{n \in \bN^*}$ has compact closure in $\tau_{\mathrm w}$.

Are there $(\mu_t)_{t \in [0, 1]} \subset \sP_2 (\bR^d)$ and a sub-sequence $(n_k)_{k \in \bN^*}$ such that for each $t \in [0, 1]$: \begin{equation} \mu^{n_k}_t \rightharpoonup \mu_t \quad \text{as} \quad k \to \infty. \end{equation}

Thank you for your elaboration.

Answer 1

Here are some ideas, but I did not have the time to check every detail.

Let $T \subset [0,1]$ be countable and dense. Using a diagonal sequence argument, one can find a subsequence such that $$ \mu_n^{n_k} \rightharpoonup \mu_t $$ for all $t \in T$. Using your continuity assumption, $$ T \ni t \mapsto \mu_t $$ should be Hölder continuous. Consequently, it should be possible to extend it to all of $[0,1]$ and to conclude with the desired convergence.