Lifted Test Functions on the Space of Square-Integrable Random Vectors and Their Translation Hessians

In the setting of Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: Standing Notation, let $L^{2}(\Omega;\mathbb{R}^{d})$ be the space of classes of square-integrable random vectors in $\mathbb{R}^{d}$, which is open in itself, so that the differential calculus in force there applies to a real-valued function $\Phi$ on it: that $\Phi$ is differentiable on $L^{2}(\Omega;\mathbb{R}^{d})$, its gradient $D\Phi(X)\in L^{2}(\Omega;\mathbb{R}^{d})$ at a point $X$, and the class $C^{1}(L^{2}(\Omega;\mathbb{R}^{d}))$, are as defined there. For $a\in\mathbb{R}^{d}$, $c_{a}$ is the constant class with value $a$; the origin $0_{\mathbb{R}^{d}}$ and the set $\mathcal{S}(d)$ of symmetric real $d\times d$ matrices are those of Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: Standing Notation §matrices; being twice continuously differentiable along translations at a point and twice continuously differentiable along translations are as defined there; the class $C^{2}$ on $\mathbb{R}^{d}$ and the Hessian matrix $D^{2}\phi(a)$ of a function $\phi$ of that class are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives; and Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian is read with $m=d$.

1. (Lifted test function)¶ A function $\Phi:L^{2}(\Omega;\mathbb{R}^{d})\to\mathbb{R}$ is a lifted test function if it has the following two properties.

(a) $\Phi$ belongs to $C^{1}(L^{2}(\Omega;\mathbb{R}^{d}))$.

(b) $\Phi$ is twice continuously differentiable along translations.

2. (The translation Hessian)¶ Let $\Phi$ be a lifted test function, let $X\in L^{2}(\Omega;\mathbb{R}^{d})$, and let $\phi_{X}:\mathbb{R}^{d}\to\mathbb{R}$ be the function $\phi_{X}(a)=\Phi(X+c_{a})$, which is of class $C^{2}$ on $\mathbb{R}^{d}$ by property (b), that is by The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors §on-space together with The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors §translations. The translation Hessian of $\Phi$ at $X$ is

an element of $\mathcal{S}(d)$ by Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian §hessian.

Law-invariance is not among the requirements, so $D\Phi(X)$ is not asserted to be of the form $\eta\circ X$ for a vector field $\eta$ over the law of $X$, as property (b) of Test Functions on the Wasserstein Space: the Intrinsic Gradient and the Translation Hessian §test demands of the lift of a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$; and neither $D\Phi(X)$ nor $H_{\Phi}(X)$ is asserted to depend on $X$ only through its law.