Noise Intrinsic Test Functions on the Noise Wasserstein Space

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $Q$ be a subset of the set $\mathcal{P}^{a}_{\rho}$ of The Measures Noise-Connected to the Reference Measure §space, and let $\varphi:\mathcal{P}^{a}_{\rho}\to\mathbb{R}$. Continuity of $\varphi$ is understood between the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ of The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §metric and $\mathbb{R}$ with the absolute-value metric. Differentiability along noise couplings and the gradient along noise couplings $\nabla\varphi(\mu)\in L^{2}(\mu;X^{a})$ are those of that definition, $T^{a}_{\mu}\subseteq L^{2}(\mu;X^{a})$ is the noise tangent space at $\mu$, couplings of vanishing noise cost and strong convergence of noise fields along them are those of that definition, and sequences are those of that definition.

(Noise intrinsic test function)¶ The function $\varphi$ is a noise intrinsic test function on $Q$ if it has the following three properties.

(a)¶ $\varphi$ is continuous on $\mathcal{P}^{a}_{\rho}$.

(b)¶ For every $\mu\in Q$, $\varphi$ is differentiable along noise couplings at $\mu$ and $\nabla\varphi(\mu)\in T^{a}_{\mu}$.

(c)¶ Let $\mu\in Q$, let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $Q$, and let $(\pi_{n})_{n\in\mathbb{N}}$ be a sequence of couplings of vanishing noise cost from $(\mu_{n})_{n\in\mathbb{N}}$ to $\mu$. Then the sequence $(\nabla\varphi(\mu_{n}))_{n\in\mathbb{N}}$, the gradients being those of property (b), converges strongly to $\nabla\varphi(\mu)$ along $(\pi_{n})_{n\in\mathbb{N}}$.