The Langevin Free-Energy Pair of a Confining Potential on the Wasserstein Space

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, for $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ let $L^{2}(\mu;\mathbb{R}^{d})$ be the space of square-integrable vector fields and $T_{\mu}$ the tangent space, a linear subspace of $L^{2}(\mu;\mathbb{R}^{d})$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §closed. Finite entropy, the entropy $\mathrm{Ent}$ and the set $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ are those of that definition; finite Fisher information and the set $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ are those of that definition, and the score of $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ is $\xi_{\mu}\in T_{\mu}$; integrable is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. Let $V$ be a confining potential on $\mathbb{R}^{d}$, with gradient $DV(x)$ and gradient map $\nabla V:x\mapsto DV(x)$, which is Borel by Basic Properties of a Confining Potential on Euclidean Space: Continuity, Quadratic Minorants, the Tangent Inequality, Hessian Bounds, Growth under Translation, and Integrability §regularity; and let $\sigma\in\mathbb{R}$ be positive, with $\sigma^{2}=\sigma\sigma$ and $\tfrac{\sigma^{2}}{2}$ the product of $\sigma^{2}$ with the multiplicative inverse of $2=1+1$, which exists by claim 8 of Elementary Order Arithmetic in an Ordered Field. The letter $\sigma$ denotes the noise intensity; the swap map written $\sigma$ in Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §swap is not used.

(The Langevin free-energy pair)¶ The Langevin free-energy pair with potential $V$ and noise intensity $\sigma$ is the quadruple $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ given as follows. The set $\mathcal{D}$ consists of the $\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ for which $V$ is integrable with respect to $\mu$, and

The set $\mathcal{D}_{\Sigma}$ consists of the $\mu\in\mathcal{D}\cap\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ with $\int_{\mathbb{R}^{d}}\lVert\nabla V\rVert^{2}\,d\mu<\infty$. For such $\mu$ the class of $\nabla V$ in $L^{2}(\mu;\mathbb{R}^{d})$, again written $\nabla V$, belongs to $T_{\mu}$ by The Gradient of a Convex Continuously Differentiable Function Belongs to the Tangent Space When It Is Square-Integrable §tangent, $V$ being convex and of class $C^{1}$ on $\mathbb{R}^{d}$ (clause 2 of C^k Maps on a Euclidean Open Set); and

the combination being formed in the linear subspace $T_{\mu}$.