The Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation with a Linear Running Cost: Its Viscosity Solution Is the Integral of the Pointwise Solution, Which It Determines

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, and in the setting of Second-Order Equations on Euclidean Open Sets, whose clauses are used with the dimension written $q$ there equal to $d$, and where The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space speaks of $n$ and $U$ it is applied with $n=d$ and $U=\mathbb{R}^{d}$; otherwise the letter $q$ denotes a vector field, as in The Intrinsic Calculus on the Wasserstein Space: Standing Notation §operators, and $\mu$ always denotes a probability measure, never a scalar. Let $c$ be a variance vector, with weighted square $|x|_{c}^{2}$ ($x\in\mathbb{R}^{d}$) as in The Diagonal Gaussian Density on Euclidean Space and Its Notation §scaling, let $a\in\mathbb{R}$ be positive, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian free-energy pair with variances $c$ and temperature $a$. Let $\lambda_{0},\theta\in\mathbb{R}$ satisfy $0<\lambda_{0}$ and $0<\theta\le1$, let $\tilde{g}:\mathbb{R}^{d}\to\mathbb{R}$ be bounded and uniformly continuous (for the Euclidean distance and the metric of The Absolute Value Metric on the Real Line), and fix $C\in\mathbb{R}$ with $|\tilde{g}(x)|\le C$ for every $x\in\mathbb{R}^{d}$. By Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral, applied with $m=d$, $\tilde{g}$ is Borel and integrable with respect to every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, and the function $G:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$, $G(\mu)=\int_{\mathbb{R}^{d}}\tilde{g}\,d\mu$, is uniformly continuous from $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ to $\mathbb{R}$ and satisfies $|G(\mu)|\le C$. Let $V_{c}:\mathbb{R}^{d}\to\mathbb{R}$, $V_{c}(x)=\tfrac{a}{2}|x|_{c}^{2}$, which is of class $C^{2}$ on $\mathbb{R}^{d}$ by Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic, applied to the diagonal matrix with diagonal entries $a/c_{1},\dots,a/c_{d}$, linear coefficient $0$ and constant term $0$, and let $F$ be the penalty-drift Hamilton-Jacobi operator on $\mathbb{R}^{d}$ with potential $V_{c}$, discount $\lambda_{0}$, control cost $\theta$, noise intensity $2a$ and running cost $\tilde{g}$, with viscosity sub- and supersolutions on $\mathbb{R}^{d}$ as in The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space §equation. Viscosity solutions of the lifted Ornstein-Uhlenbeck Hamilton-Jacobi equation are taken with discount $\lambda_{0}$, control cost $\theta$ and running cost $G$, and $\lambda_{0}^{-1}$ is the multiplicative inverse of $\lambda_{0}$.

1. (The pointwise solution)¶ There is exactly one bounded function $w:\mathbb{R}^{d}\to\mathbb{R}$ that is both a viscosity subsolution and a viscosity supersolution of $F$ on $\mathbb{R}^{d}$. It is continuous and satisfies $|w(x)|\le\lambda_{0}^{-1}C$ for every $x\in\mathbb{R}^{d}$; in particular it is Borel and integrable with respect to every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$.

2. (Representation)¶ With $w$ as in clause 1, the function $\mathcal{U}:\mathcal{D}\to\mathbb{R}$, $\mathcal{U}(\mu)=\int_{\mathbb{R}^{d}}w\,d\mu$, is a viscosity solution of the lifted Ornstein-Uhlenbeck Hamilton-Jacobi equation, and every bounded viscosity solution $u:\mathcal{D}\to\mathbb{R}$ of that equation satisfies $u(\mu)=\mathcal{U}(\mu)$ for every $\mu\in\mathcal{D}$.

3. (Recovery of the pointwise solution)¶ For positive $\varepsilon\in\mathbb{R}$ let $\gamma_{\varepsilon}$ be the diagonal Gaussian measure with the variance vector $(\varepsilon,\dots,\varepsilon)\in\mathbb{R}^{d}$, and for $x\in\mathbb{R}^{d}$ let $\eta_{x,\varepsilon}$ be its image under the translation $\mathbb{R}^{d}\to\mathbb{R}^{d}$, $y\mapsto x+y$. Then $\eta_{x,\varepsilon}\in\mathcal{D}$ for all such $x$ and $\varepsilon$, and for every $x\in\mathbb{R}^{d}$ and every sequence $(\varepsilon_{j})_{j\in\mathbb{N}}$ of positive real numbers converging to $0$, the sequence $(\mathcal{U}(\eta_{x,\varepsilon_{j}}))_{j\in\mathbb{N}}$ converges to $w(x)$.