Lambda-Displacement Convexity of a Noise Penalty Pair

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 $\mathcal{P}^{a}_{\rho}$ and let $\lambda\in\mathbb{R}$. For $\mu\in\mathcal{D}_{\Sigma}$ one has $\mu\in\mathcal{D}$, so that $\mathcal{E}(\mu)$ is a real number, and the score $\Sigma(\mu)$ lies in $T^{a}_{\mu}$, hence in $L^{2}(\mu;X^{a})$, both by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair. For $\nu\in\mathcal{D}$ the ordered pair $(\mu,\nu)$ is noise-connected by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §connected, since $\mu,\nu\in\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, so that noise-optimal couplings $\pi\in\Pi^{a}(\mu,\nu)$ are defined, and for such $\pi$ the noise displacement pairing $\mathcal{J}^{a}(\Sigma(\mu),\pi)$ and the noise cost $I^{a}(\pi)$ are the real numbers of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §couplings.

($\lambda$-displacement convexity)¶ The noise penalty pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is $\lambda$-displacement convex if

$$\mathcal{E}(\mu)+\mathcal{J}^{a}(\Sigma(\mu),\pi)+\tfrac{\lambda}{2}\,I^{a}(\pi)\ \le\ \mathcal{E}(\nu)$$

for every $\mu\in\mathcal{D}_{\Sigma}$, every $\nu\in\mathcal{D}$ and every noise-optimal coupling $\pi\in\Pi^{a}(\mu,\nu)$. ¶ It is displacement convex if it is $0$-displacement convex.