Talagrand, HWI and Log-Sobolev Inequalities for a Uniformly Displacement Convex Noise Penalty Pair with a Point of Zero Score

In the setting of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a noise penalty pair on $\mathcal{P}^{a}_{\rho}$, let $\lambda\in\mathbb{R}$ be positive, and suppose that the pair is $\lambda$-displacement convex. Let $\mu_{*}\in\mathcal{D}_{\Sigma}$ be such that $\Sigma(\mu_{*})$ is the zero vector of $L^{2}(\mu_{*};X^{a})$. For $\mu\in\mathcal{D}_{\Sigma}$, $\Sigma(\mu)\in L^{2}(\mu;X^{a})$ by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, with norm $\lVert\Sigma(\mu)\rVert_{\mu}$; and $W_{a}$ is the noise Wasserstein distance of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §space, defined on $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$.

1. (Talagrand inequality)¶ For every $\nu\in\mathcal{D}$,

$$\mathcal{E}(\mu_{*})+\frac{\lambda}{2}\,W_{a}(\mu_{*},\nu)^{2}\le\mathcal{E}(\nu);$$

in particular $\mathcal{E}(\mu_{*})\le\mathcal{E}(\nu)$ for every $\nu\in\mathcal{D}$.

2. (HWI inequality)¶ For every $\mu\in\mathcal{D}_{\Sigma}$,

$$\mathcal{E}(\mu)-\mathcal{E}(\mu_{*})\le\lVert\Sigma(\mu)\rVert_{\mu}\,W_{a}(\mu,\mu_{*})-\frac{\lambda}{2}\,W_{a}(\mu,\mu_{*})^{2}.$$

3. (Log-Sobolev inequality)¶ For every $\mu\in\mathcal{D}_{\Sigma}$,

$$\mathcal{E}(\mu)-\mathcal{E}(\mu_{*})\le\frac{1}{2\lambda}\,\lVert\Sigma(\mu)\rVert_{\mu}^{2}.$$