Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $\varphi:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$. Differentiability along couplings and the gradient along couplings $\nabla\varphi(\mu)$ are those of that definition, and intrinsic test functions on a subset of $\mathcal{P}(\mathbb{T}^{d})$ are those of that definition. For a map $g:\mathcal{P}(\mathbb{T}^{d})\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ and $\mu\in\mathcal{P}(\mathbb{T}^{d})$, $g(\mu,\cdot)$ denotes the map $x\mapsto g(\mu,x)$, $g_{i}$ its $i$th component, and $\partial_{j}g_{i}(\mu,x)$ the partial derivative of $g_{i}(\mu,\cdot)$ at $x$ when it exists. If each $g_{i}(\mu,\cdot)$ belongs to $C_{\mathrm{per}}$, then $g(\mu,\cdot)$ is Borel and bounded by The Torus Wasserstein Distance: Comparison with the Euclidean Distance, Wrapping, and Integrals of Periodic Functions §integrals, so it has a class in $L^{2}(\mu;\mathbb{R}^{d})$.

1. (Gradient field)¶ A map $g:\mathcal{P}(\mathbb{T}^{d})\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a gradient field of $\varphi$ if the following hold.

(a)¶ For every $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and $i\in[d]$, $g_{i}(\mu,\cdot)\in C^{1}_{\mathrm{per}}$.

(b)¶ For every $\mu\in\mathcal{P}(\mathbb{T}^{d})$, $\varphi$ is differentiable along couplings at $\mu$ and $\nabla\varphi(\mu)$ is the class of $g(\mu,\cdot)$ in $L^{2}(\mu;\mathbb{R}^{d})$.

(c)¶ Whenever $(\mu_{n})_{n\in\mathbb{N}}$ converges to $\mu$ in $(\mathcal{P}(\mathbb{T}^{d}),W_{\mathbb{T}})$ and $(x_{n})_{n\in\mathbb{N}}$ converges to $x$ in $\mathbb{R}^{d}$, the real sequences $(g_{i}(\mu_{n},x_{n}))_{n\in\mathbb{N}}$ and $(\partial_{j}g_{i}(\mu_{n},x_{n}))_{n\in\mathbb{N}}$ converge to $g_{i}(\mu,x)$ and $\partial_{j}g_{i}(\mu,x)$, for all $i,j\in[d]$.

2. (Laplacian test function)¶ $\varphi$ is a Laplacian test function if it is an intrinsic test function on $\mathcal{P}(\mathbb{T}^{d})$ and has a gradient field.