First- and Second-Order Conditions at a Penalised Extremum of an Intrinsic Test Function on the Wasserstein Space

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\delta\in\mathbb{R}$ be positive, let $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\chi$ be an intrinsic test function on $Q$, with gradient along couplings $\nabla\chi(\mu)$ at $\mu\in Q$ and translation Hessian $H_{\chi}(\mu)$. The functions $\chi-\delta\mathcal{E}$ and $\chi+\delta\mathcal{E}$ on $\mathcal{D}$ take the values $\chi(\mu)-\delta\,\mathcal{E}(\mu)$ and $\chi(\mu)+\delta\,\mathcal{E}(\mu)$ at $\mu\in\mathcal{D}$, and local maxima and local minima relative to $\mathcal{D}$ are taken in the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$. The set $\mathcal{S}(d)$ of symmetric real $d\times d$ matrices and its order $\preceq$ are those fixed there; for $\mu\in\mathcal{D}$, $H_{\mathcal{E}}(\mu)\in\mathcal{S}(d)$ is the translation Hessian of the penalty at $\mu$, and $\delta H_{\mathcal{E}}(\mu)$ and $-\delta H_{\mathcal{E}}(\mu)$ are its real multiples by $\delta$ and by $-\delta$, which lie in $\mathcal{S}(d)$ by claim 1 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure. For $\mu\in Q\cap\mathcal{D}_{\Sigma}$ both $\nabla\chi(\mu)$ and the score $\Sigma(\mu)$ lie in $T_{\mu}$, by property (b) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians (i) and by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (i). Then the following hold.

1. (Penalised maximum)(i) Let $\hat{\mu}\in Q\cap\mathcal{D}_{\Sigma}$ be a point at which the function $\chi-\delta\mathcal{E}$ has a local maximum relative to $\mathcal{D}$. Then

$$\nabla\chi(\hat{\mu})=\delta\,\Sigma(\hat{\mu}),\qquad H_{\chi}(\hat{\mu})\preceq\delta H_{\mathcal{E}}(\hat{\mu}).$$

2. (Penalised minimum)(ii) Let $\hat{\mu}\in Q\cap\mathcal{D}_{\Sigma}$ be a point at which the function $\chi+\delta\mathcal{E}$ has a local minimum relative to $\mathcal{D}$. Then

$$\nabla\chi(\hat{\mu})=-\delta\,\Sigma(\hat{\mu}),\qquad -\delta H_{\mathcal{E}}(\hat{\mu})\preceq H_{\chi}(\hat{\mu}).$$