Bounding $2$-Wasserstein distance and the $L^1$ distance

Question

My questions come from the paper Logarithmic Sobolev inequalities for some nonlinear PDE’s written by F. Malrieu (May 2001) where author omitted a good amount of details to be filled. Suppose that $W$ is convex, even, with polynomial growth and $U$ is uniformly convex. Let $\mu_N$ to be the probability measure with density $$\mu_N = \frac{1}{Z_N}\,\exp\left(-\sum_{i=1}^N U(x_i) - \frac{1}{2N}\sum_{i,j=1}^N W(x_i-x_j)\right)$$ with $Z_N$ being a normalization constant rendering $\mu_N$ to be a probability density function, also let $\bar{u}$ be the unique minimizer (stationary measure) of the free energy functional defined by $$\eta(f) = \int f(x)\,\log f(x)\,\mathrm{d} x + \int U(x)\,f(x)\,\mathrm{d} x + \frac 12 \iint W(x-y)\,f(x)\,f(y)\,\mathrm{d}x\,\mathrm{d}y,$$ i.e., $\bar{u}$ is the unique solution of $$ \bar{u}(x) = \frac{1}{Z}\,\exp\left(U(x) - W*\bar{u}(x)\right) $$ with $Z = \int \exp\left(U(x) - W*\bar{u}(x)\right)\,\mathrm{d}x$. It is implicitly used in pp. 15 and pp.16 of the aforementioned paper that we expect to have $$\|\mu_{1,N} - \bar{u}\|_1 \leq \frac{K}{\sqrt{N}} \quad \textrm{and} \quad \mathrm{W}_2(\mu_{1,N},\bar{u}) \leq \frac{K}{\sqrt{N}}, \tag{a}$$ where $\mu_{1,N}$ represents the first marginal of $\mu_N$ and $K > 0$ is some fixed constant. Can anyone help me to figure out the claimed bounds on the $L^1$ distance and the $2$-Wasserstein distance ?

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Remark: I personally do not think the proof of the first bound in (a) automatically yields the second inequality in (a)

Answer 1

Referring to the proof of Prop 3.21 in Malrieu 2001, after the triangle inequality is applied twice, two of the terms are bounded via the upper bound $$ W_2(u_t,u_t^{(1,N)}) \vee W_2(\mu_{1,N},\bar{u}) \le \sup_{s \ge 0} \sqrt{E|X_s^{1,N}-\bar{X}_s^1|^2} \tag{$\star$}$$ which accounts for the factor $2$. The reason this bound holds is because of the supremum over $s$, which indicates that it holds for all $s\ge0$ including $s=t$ and $s=\infty$ which gives (via the coupling characterization of the 2 Wasserstein distance) the upper bounds in ($\star$) on $W_2(u_t,u_t^{(1,N)})$ and $W_2(\mu_{1,N},\bar{u}) $, respectively. To finish, Theorem 3.3 is invoked.

For the related $L^1$ bound between the densities, one uses the analogous upper bound $$ \|u_t-u_t^{(1,N)}\|_1 \vee \|\mu_{1,N}-\bar{u}\|_1 \le \sup_{s \ge 0} \|u_s-u_s^{(1,N)}\|_1 \;. $$ To finish, Prop. 3.13 is invoked with $k=1$.