The Potential Energy of a Confining Potential: Smoothness in Translations, First Variation, Closed Sublevel Sets and Convergence at Bounded Energy

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, with the Wasserstein space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ and the test functions $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ with their gradient maps $\nabla\psi$ as fixed there. Integrable is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures; limits of real sequences are those of that definition; being of class $C^{2}$ on $\mathbb{R}^{d}$, and the partial derivatives $\partial_{i}$, are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, $\mathbb{R}^{d}$ being open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous; differentiability at $0$ of a function $\mathbb{R}\to\mathbb{R}$ is that of Derivative at an Interior Point; and continuity of a function $\mathbb{R}^{d}\to\mathbb{R}$ refers to the Euclidean distance and the metric of The Absolute Value Metric on the Real Line. Let $V$ be a confining potential on $\mathbb{R}^{d}$, with partial derivatives $\partial_{i}V$ and $\partial_{j}\partial_{i}V$ ($i,j\in[d]$), gradient $DV(x)$ at $x$ and gradient map $\nabla V:x\mapsto DV(x)$; the functions $V$, $\partial_{i}V$ and $\partial_{j}\partial_{i}V$ are continuous and Borel, and $\nabla V$ is Borel, by Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §regularity.

1. (Translations)¶ Let $\mu\in\mathcal{P}(\mathbb{R}^{d})$ be such that $V$ is integrable with respect to $\mu$. Then the function

is of class $C^{2}$ on $\mathbb{R}^{d}$, and for all $i,j\in[d]$ the function $\partial_{j}\partial_{i}V$ is integrable with respect to $\mu$ and $\partial_{j}\partial_{i}\Phi_{\mu}(0_{\mathbb{R}^{d}})=\int_{\mathbb{R}^{d}}\partial_{j}\partial_{i}V\,d\mu$.

2. (First variation)¶ Let $\mu$ be as in claim 1 and let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$. Then for every $t\in\mathbb{R}$ the function $x\mapsto V(x+t\,\nabla\psi(x))$ is Borel and integrable with respect to $\mu$, the function $\nabla V\cdot\nabla\psi$ is Borel and integrable with respect to $\mu$, and the function

is differentiable at $0$ with derivative $\int_{\mathbb{R}^{d}}\nabla V\cdot\nabla\psi\,d\mu$.

3. (Closed sublevel sets)¶ Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\mu_{n}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ for $n\in\mathbb{N}$ satisfy $\lim_{n\to\infty}W_{2}(\mu_{n},\mu)=0$, and let $c\in\mathbb{R}$ be such that for every $n\in\mathbb{N}$, $V$ is integrable with respect to $\mu_{n}$ and $\int_{\mathbb{R}^{d}}V\,d\mu_{n}\le c$. Then $V$ is integrable with respect to $\mu$, $\int_{\mathbb{R}^{d}}V\,d\mu\le c$, and for every positive $\varepsilon\in\mathbb{R}$ there is $N\in\mathbb{N}$ with $\int_{\mathbb{R}^{d}}V\,d\mu-\varepsilon<\int_{\mathbb{R}^{d}}V\,d\mu_{n}$ for every $n\ge N$.

4. (Convergence at bounded potential energy)¶ Let $\mu$, $(\mu_{n})_{n\in\mathbb{N}}$ and $c$ be as in claim 3, and let $h:\mathbb{R}^{d}\to\mathbb{R}$ be continuous and such that for every positive $\varepsilon\in\mathbb{R}$ there is $C_{\varepsilon}\in\mathbb{R}$ with $|h(x)|\le\varepsilon|V(x)|+C_{\varepsilon}$ for every $x\in\mathbb{R}^{d}$. Then $h$ is integrable with respect to $\mu$ and to every $\mu_{n}$, and