Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry

In the settings of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, with the coordinates $x_{k}$ and the maps $p_{n}:X\to\mathbb{R}^{n}$ of clause A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §background of the first, the noise weights $a$, the noise space $X^{a}$ with norm $|\cdot|_{a}$ and orthonormal basis $(f_{k})_{k\in\mathbb{N}}$, and the orthonormal basis $(e_{k})_{k\in\mathbb{N}}$ of $X$, let $d\in\mathbb{N}$; $\mathbb{R}^{d}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. Let $v:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}^{d}$ in the sense of C^k Maps on a Euclidean Open Set, with partial derivatives $\partial_{k}v$ and $\partial_{j}\partial_{k}v$ ($j,k\in[d]$) as in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives. Let $K\in\mathbb{R}$ be nonnegative.

(Admissible cylindrical potential)¶ The function $V=v\circ p_{d}:X\to\mathbb{R}$ is an admissible cylindrical potential with head dimension $d$, profile $v$ and semiconvexity constant $K$ in the noise norm if the following four conditions hold. Dependents say ``an admissible cylindrical potential $V=v\circ p_{d}$'', meaning that the datum $(d,v,K)$ is fixed with it.

(a) (Lower bound)¶ There is $b\in\mathbb{R}$ with $-b\le v(u)$ for every $u\in\mathbb{R}^{d}$.

(b) (Semiconvexity in the noise norm)¶ The Hessian of $v$ is bounded below by $-K$ times the noise quadratic form: for all $u,h\in\mathbb{R}^{d}$,

$$\sum_{j=1}^{d}\sum_{k=1}^{d}\partial_{j}\partial_{k}v(u)\,h_{j}h_{k}\ \ge\ -K\sum_{k=1}^{d}\frac{h_{k}^{2}}{a_{k}} .$$

(c) (Slope bound)¶ There is $C\in\mathbb{R}$ with

$$\sum_{k=1}^{d}a_{k}\bigl(\partial_{k}v(u)\bigr)^{2}\ \le\ C\,\bigl(1+|v(u)|\bigr)^{2}\qquad\text{for every }u\in\mathbb{R}^{d}.$$

(d) (Curvature small against the slope)¶ For every positive $\varepsilon\in\mathbb{R}$ there is $C_{\varepsilon}\in\mathbb{R}$ with

$$\sum_{k=1}^{d}a_{k}\,\partial_{k}\partial_{k}v(u)\ \le\ \varepsilon\sum_{k=1}^{d}a_{k}\bigl(\partial_{k}v(u)\bigr)^{2}+C_{\varepsilon}\Bigl(1+\sum_{k=1}^{d}u_{k}^{2}\Bigr)\qquad\text{for every }u\in\mathbb{R}^{d}.$$

(Noise gradient)¶ For such $V$, the noise gradient of $V$ is the map $\nabla_{a}V:X\to X^{a}$,

$$\nabla_{a}V(x)=\sum_{k=1}^{d}a_{k}\,\partial_{k}v\bigl(p_{d}(x)\bigr)\,e_{k}\qquad(x\in X),$$

a finite linear combination of the vectors $e_{k}$; $X^{a}$ is a linear subspace of $X$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert and contains each $e_{k}=a_{k}^{-1/2}f_{k}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis, so $\nabla_{a}V(x)\in X^{a}$. For $k\in\mathbb{N}$ write $\partial_{k}V(x)=\partial_{k}v(p_{d}(x))$ if $k\le d$ and $\partial_{k}V(x)=0$ if $k>d$. Here $\nabla_{a}V$ and $\partial_{k}V$ are notations for the displayed functions.