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

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, assume that $(\Omega,\mathcal{F},P)$ is rich, 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 and let $\chi$ be a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$, with intrinsic gradient $\nabla\chi(\mu)$ and translation Hessian $H_{\chi}(\mu)$ at $\mu$. 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 $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (i). For $\mu\in\mathcal{D}_{\Sigma}$ the score $\Sigma(\mu)$ and the intrinsic gradient $\nabla\chi(\mu)$ both belong to the tangent space $T_{\mu}$, a subset of $L^{2}(\mu;\mathbb{R}^{d})$, in which $\delta\,\Sigma(\mu)$ and $-\delta\,\Sigma(\mu)$ are real multiples. The set $\mathcal{S}(d)$ of symmetric real $d\times d$ matrices, with the zero matrix $0_{d}$, and its order $\preceq$ are those fixed there. In this statement test function means a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\nabla\chi(\mu)$ its intrinsic gradient; the test functions $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (iii) keep their notation. Then the following hold.

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

$$\nabla\chi(\hat{\mu})=\delta\,\Sigma(\hat{\mu}),\qquad H_{\chi}(\hat{\mu})\preceq0_{d}.$$

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

$$\nabla\chi(\hat{\mu})=-\delta\,\Sigma(\hat{\mu}),\qquad 0_{d}\preceq H_{\chi}(\hat{\mu}).$$