Penalised Extrema of the Integral of a Lipschitz Semiconvex Function Relative to the Gaussian Free-Energy Pair: Finite Relative Fisher Information and the First-Order Condition

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $c$ be a variance vector, let $a\in\mathbb{R}$ be positive, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian free-energy pair with variances $c$ and temperature $a$. Let $L,K\in\mathbb{R}$ be nonnegative, let $w:\mathbb{R}^{d}\to\mathbb{R}$ be bounded and Lipschitz with constant $L$ (for the Euclidean distance and the metric of The Absolute Value Metric on the Real Numbers), and let $\nabla w$ be the Borel map of The Almost-Everywhere Gradient of a Lipschitz Semiconvex or Semiconcave Function Is Tangent at Every Absolutely Continuous Measure §borel, which satisfies $\lVert\nabla w(x)\rVert\le L$ for every $x\in\mathbb{R}^{d}$, so that its class lies in $L^{2}(\mu;\mathbb{R}^{d})$ for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. The function $w$ is uniformly continuous by A Lipschitz Map is Uniformly Continuous and Continuous, so by Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral, applied with $m=d$, it is Borel and integrable with respect to every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$; let $W:\mathcal{D}\to\mathbb{R}$, $W(\mu)=\int_{\mathbb{R}^{d}}w\,d\mu$. Let $\delta\in\mathbb{R}$ be positive and let $\varphi$ be an intrinsic test function on $\mathcal{D}$, with gradients along couplings $\nabla\varphi(\mu)\in T_{\mu}$. Local maxima and local minima relative to $\mathcal{D}$ are understood in the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$; semiconvexity is on $\mathbb{R}^{d}$, a convex set directly from that definition, and $-w$ has the value $-w(x)$ at $x$.

1. (Penalised maxima)¶ Suppose that $w$ is semiconvex with constant $K$, and let $\hat{\mu}\in\mathcal{D}$ be a point at which the function $\mathcal{D}\to\mathbb{R}$ with value $W(\mu)-\delta\,\mathcal{E}(\mu)-\varphi(\mu)$ at $\mu$ has a local maximum relative to $\mathcal{D}$. Then $\hat{\mu}\in\mathcal{D}_{\Sigma}$ and

$$\nabla w=\nabla\varphi(\hat{\mu})+\delta\,\Sigma(\hat{\mu})\qquad\text{in }L^{2}(\hat{\mu};\mathbb{R}^{d}).$$

2. (Penalised minima)¶ Suppose that $-w$ is semiconvex with constant $K$, and let $\hat{\mu}\in\mathcal{D}$ be a point at which the function $\mathcal{D}\to\mathbb{R}$ with value $W(\mu)+\delta\,\mathcal{E}(\mu)-\varphi(\mu)$ at $\mu$ has a local minimum relative to $\mathcal{D}$. Then $\hat{\mu}\in\mathcal{D}_{\Sigma}$ and

$$\nabla w=\nabla\varphi(\hat{\mu})-\delta\,\Sigma(\hat{\mu})\qquad\text{in }L^{2}(\hat{\mu};\mathbb{R}^{d}).$$