Noise Penalty Pairs with Regular Penalised Maxima

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 $\mathcal{P}^{a}_{\rho}$. Noise intrinsic test functions on $\mathcal{D}$ are those of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §gradients, and local maxima relative to $\mathcal{D}$ are taken in the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §space. For $\chi:\mathcal{P}^{a}_{\rho}\to\mathbb{R}$ and positive $\lambda\in\mathbb{R}$, $\chi-\lambda\mathcal{E}$ is the function on $\mathcal{D}$ with value $\chi(\mu)-\lambda\,\mathcal{E}(\mu)$ at $\mu$.

(Regular penalised maxima)¶ The noise penalty pair has regular penalised maxima if for every noise intrinsic test function $\chi$ on $\mathcal{D}$ and every positive $\lambda\in\mathbb{R}$, every $\mu\in\mathcal{D}$ at which $\chi-\lambda\mathcal{E}$ has a local maximum relative to $\mathcal{D}$ belongs to $\mathcal{D}_{\Sigma}$.