The First-Order Condition at a Penalised Extremum of a Noise Intrinsic Test Function on the Noise Wasserstein Space

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 $\delta\in\mathbb{R}$ be positive, let $Q\subseteq\mathcal{P}^{a}_{\rho}$ and let $\chi:\mathcal{P}^{a}_{\rho}\to\mathbb{R}$ be a noise intrinsic test function on $Q$, with gradient along noise couplings $\nabla\chi(\mu)\in L^{2}(\mu;X^{a})$ at $\mu\in Q$. 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 $(\mathcal{P}^{a}_{\rho},W_{a})$ of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §space. For $\mu\in Q\cap\mathcal{D}_{\Sigma}$ both $\nabla\chi(\mu)$ and the score $\Sigma(\mu)$ lie in the noise tangent space $T^{a}_{\mu}\subseteq L^{2}(\mu;X^{a})$, by property (b) of Noise Intrinsic Test Functions on the Noise Wasserstein Space §test and by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair. Then the following hold.

1. (Penalised maximum)¶ 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

$$\nabla\chi(\hat{\mu})=\delta\,\Sigma(\hat{\mu})\qquad\text{in }L^{2}(\hat{\mu};X^{a}).$$

2. (Penalised minimum)¶ 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

$$\nabla\chi(\hat{\mu})=-\delta\,\Sigma(\hat{\mu})\qquad\text{in }L^{2}(\hat{\mu};X^{a}).$$