Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, let $V=v\circ p_{d}$ be an admissible cylindrical potential with head dimension $d$, profile $v$ and semiconvexity constant $K$ in the noise norm, with noise gradient $\nabla_{a}V$ and the functions $\partial_{k}V$ of that clause, and let $b$ and $C$ be constants as in Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §below and Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §slope. $\langle\cdot,\cdot\rangle_{a}$ is the inner product of $X^{a}$; for $x\in X$ write $|p_{d}(x)|_{a}^{2}=\sum_{k=1}^{d}x_{k}^{2}/a_{k}$, a nonnegative real number, with $|p_{d}(x)|_{a}$ its nonnegative square root; $\exp$ is the exponential function of Basic Properties of the Exponential Function. For $\mu\in\mathcal{P}(X)$, $L^{2}(\mu;X^{a})$ and $T^{a}_{\mu}$ are the space of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields and the noise tangent space.

1. (Continuity and the noise gradient)¶ $V$ is continuous on $X$, hence Borel, and $-b\le V(x)$ for every $x\in X$. The map $\nabla_{a}V:X\to X^{a}$ is continuous, hence measurable into $X^{a}$, with $|\nabla_{a}V(x)|_{a}^{2}=\sum_{k=1}^{d}a_{k}\,\partial_{k}V(x)^{2}$ and $\langle\nabla_{a}V(x),h\rangle_{a}=\sum_{k=1}^{d}\partial_{k}V(x)\,h_{k}$ for $x\in X$ and $h\in X^{a}$ with coordinates $h_{k}$. For every $k\in\mathbb{N}$ the function $\partial_{k}V$ is continuous and is the $k$-th partial derivative of $V$, $V$ being differentiable on $X$.

2. (Growth under noise translations)¶ The constant $C$ of Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §slope is nonnegative, since there the left-hand side is nonnegative and $\bigl(1+|v(u)|\bigr)^{2}$ is positive. Put $L=C^{1/2}\bigl(|b+1|+2\bigr)$, with $C^{1/2}$ the nonnegative square root of $C$; so $L$ is nonnegative. Then

$$V(x+h)+b+1\le\bigl(V(x)+b+1\bigr)\exp\bigl(L\,|h|_{a}\bigr)\qquad\text{for all }x\in X\text{ and }h\in X^{a}.$$

3. (Exponential growth)¶ With this $L$, $V(x)+b+1\le\bigl(v(0)+b+1\bigr)\exp\bigl(L\,|p_{d}(x)|_{a}\bigr)$ for every $x\in X$.

4. (Integrable slope)¶ Let $\mu\in\mathcal{P}(X)$ be such that $V$ is integrable with respect to $\mu$. Then $|\nabla_{a}V|_{a}$ and each $\partial_{k}V$ are integrable with respect to $\mu$.

5. (Gaussian integrability)¶ The functions $V$, $V^{2}$ and $|\nabla_{a}V|_{a}^{2}$ are integrable with respect to $\gamma_{c}$.

6. (Tangent inequality)¶ For all $x,y\in X$ with $y-x\in X^{a}$,

$$V(y)\ \ge\ V(x)+\langle\nabla_{a}V(x),y-x\rangle_{a}-\frac{K}{2}\,|y-x|_{a}^{2}.$$

7. (Tangency of the noise gradient)¶ Let $\mu\in\mathcal{P}(X)$ satisfy $\int_{X}|\nabla_{a}V|_{a}^{2}\,d\mu<\infty$. Then the class of $\nabla_{a}V$ lies in $L^{2}(\mu;X^{a})$ and belongs to $T^{a}_{\mu}$.