Heat Smoothing, Entropy, Fisher Information and Laplacian Test Functions on the Torus Wasserstein Space: Standing Notation

This setting fixes standing notation for heat smoothing and second-order calculus on the torus Wasserstein space. It introduces no new concept and asserts nothing beyond the identifications recorded below, each justified by the reference attached to it.

1. (Conventions)¶ The notation of Optimal Transport on the Flat Torus: Standing Notation is in force, and $\lambda_{d}$ is Lebesgue measure. The letter $s$ denotes a real number with $0<s\le\tfrac12$; $S_{s}$ is the heat semigroup and $\Theta_{s}$ the torus heat kernel at time $s$, and for $\mu\in\mathcal{P}(\mathbb{T}^{d})$, $p^{s}_{\mu}$ is the density function $p_{\mu}$ of The Torus Heat Kernel: Regularity and Bounds, the Density of a Heat-Smoothed Measure, and Its Lipschitz Dependence on the Measure §density at time $s$.

2. (Measures)¶ $\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$ is the set of absolutely continuous members of $\mathcal{P}(\mathbb{T}^{d})$; $\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ and $\mathrm{Ent}$ are as in The Entropy of Measures on the Torus: Nonnegativity, Absolute Continuity and Closed Sublevel Sets §finite-entropy-set; $\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$, the score $\xi_{\mu}$ and the torus Fisher information $\mathcal{I}$ are as in Finite Torus Fisher Information, the Torus Score and the Torus Fisher Information of a Probability Measure on the Torus §finite. For every $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and every $s$, the measure $S_{s}\mu$ belongs to $\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$, to $\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$ and to $\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$, by The Torus Heat Kernel: Regularity and Bounds, the Density of a Heat-Smoothed Measure, and Its Lipschitz Dependence on the Measure §density and Heat Smoothing on the Torus: Distance to the Identity, Contraction, a Smooth Positive Periodic Density, Finite Entropy, and Finite Fisher Information §entropy.

3. (Heat averages)¶ For $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and $F\in L^{2}(S_{s}\mu;\mathbb{R}^{d})$, $A^{\mu}_{s}F$ denotes the map $x\mapsto\int_{Q}F(y)\,\Theta_{s}(y-x)\,\lambda_{d}(dy)$ of Heat Averages of Vector Fields on the Torus: Regularity, Contraction, Tangency, the Divergence Identity and Synchronous Couplings §average, formed with any Borel representative of $F$ (the map does not depend on the choice, by that clause), and also its class in $L^{2}(\mu;\mathbb{R}^{d})$.

4. (Calculus on measures)¶ $T_{\mu}$ is the tangent space at $\mu$; differentiability along couplings and the gradient along couplings $\nabla\varphi(\mu)$ are those of Differentiability Along Couplings of a Function on the Torus Wasserstein Space, and Its Gradient; intrinsic test functions on a subset $\mathcal{K}$ of $\mathcal{P}(\mathbb{T}^{d})$ are those of Intrinsic Test Functions on the Torus Wasserstein Space; convergence along couplings is that of Convergence Along Couplings of Measures Carrying Square-Integrable Vector Fields on the Torus; gradient fields and Laplacian test functions are those of Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space; and for a Laplacian test function $\varphi$, $g_{\varphi}$ is its gradient field, unique by Laplacian Test Functions on the Torus: Uniqueness of the Gradient Field, a Criterion, and Linearity of the Test Classes §unique, and $\mathcal{L}\varphi$ its Laplacian. For $\varphi:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$, $\varphi\circ S_{s}$ is the map $\mu\mapsto\varphi(S_{s}\mu)$, and $\mathrm{Ent}\circ S_{s}$ is the map $\mu\mapsto\mathrm{Ent}(S_{s}\mu)$, real-valued by clause 2.

5. (Background)¶ The following results are in force by reference: Heat Smoothing on the Torus: Distance to the Identity, Contraction, a Smooth Positive Periodic Density, Finite Entropy, and Finite Fisher Information, The Torus Heat Kernel: Regularity and Bounds, the Density of a Heat-Smoothed Measure, and Its Lipschitz Dependence on the Measure, Heat Averages of Vector Fields on the Torus: Regularity, Contraction, Tangency, the Divergence Identity and Synchronous Couplings, Lower Semicontinuity of the Torus Fisher Information and Closedness of the Score Along Couplings, Laplacian Test Functions on the Torus: Uniqueness of the Gradient Field, a Criterion, and Linearity of the Test Classes, The Torus Tangent Space: Linearity of the Periodic Calculus, Closed Subspace, and Representation of Bounded Functionals on Gradients, The Entropy of Measures on the Torus: Nonnegativity, Absolute Continuity and Closed Sublevel Sets, Half the Squared Torus Wasserstein Distance to a Fixed Measure: the One-Sided Bound Along Couplings, Differentiability at an Absolutely Continuous Source, and the Tangent Gradient, Half the Squared Torus Wasserstein Distance to a Fixed Measure is an Intrinsic Test Function on the Absolutely Continuous Measures and McCann's Tangent Inequality on the Torus, and Monotonicity of the Score Along Optimal Maps.