Integrals of Lipschitz Semiconvex Viscosity Subsolutions and Lipschitz Semiconcave Viscosity Supersolutions of the Ornstein-Uhlenbeck Hamilton-Jacobi Equation Are Viscosity Sub- and Supersolutions of the Lifted Equation

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 $\lambda_{0},\theta\in\mathbb{R}$ be positive, 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), so that by Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral, applied with $m=d$, it is Borel and integrable with respect to every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, and let $G:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$, $G(\mu)=\int_{\mathbb{R}^{d}}\tilde{g}\,d\mu$. 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 $0$. 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 sub- and supersolutions of the lifted Ornstein-Uhlenbeck Hamilton-Jacobi equation are taken with discount $\lambda_{0}$, control cost $\theta$ and running cost $G$. Let $L,K\in\mathbb{R}$ be nonnegative, let $w:\mathbb{R}^{d}\to\mathbb{R}$ be bounded and Lipschitz with constant $L$ (for the Euclidean distance and the metric of The Absolute Value Metric on the Real Line), so that $w$ is uniformly continuous by A Lipschitz Map is Uniformly Continuous, hence Borel and integrable with respect to every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral; let $W:\mathcal{D}\to\mathbb{R}$, $W(\mu)=\int_{\mathbb{R}^{d}}w\,d\mu$. Semiconvexity is on $\mathbb{R}^{d}$, a convex set directly from that definition, and $-w$ has the value $-w(x)$ at $x$.

1. (Subsolutions)¶ If $w$ is semiconvex with constant $K$ and is a viscosity subsolution of $F$ on $\mathbb{R}^{d}$, then $W$ is a viscosity subsolution of the lifted Ornstein-Uhlenbeck Hamilton-Jacobi equation.

2. (Supersolutions)¶ If $-w$ is semiconvex with constant $K$ and $w$ is a viscosity supersolution of $F$ on $\mathbb{R}^{d}$, then $W$ is a viscosity supersolution of the lifted Ornstein-Uhlenbeck Hamilton-Jacobi equation.