Proof of 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

Each result cited below is universally quantified over the data in its own statement. Elementary arithmetic and order facts for real numbers (The Real Numbers: Standing Notation and Background §background) are used without citation. The quadruple $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is a penalty pair by The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §pair, and by The Gaussian Free-Energy Pair: Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure §pair $\mathcal{D}_{\Sigma}$ is the set of $\mu\in\mathcal{D}$ of finite Fisher information relative to $\gamma_{c}$, where $\mathcal{D}$ is the set of $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ of finite relative entropy with respect to $\gamma_{c}$. Write $F_{\mathrm{L}}$ for the lifted Ornstein-Uhlenbeck Hamilton-Jacobi operator with discount $\lambda_{0}$, control cost $\theta$ and running cost $G$, an operator over $\mathcal{D}_{\Sigma}$ with

$$F_{\mathrm{L}}(\nu,r,q,Y)=\lambda_{0}r+\tfrac{\theta}{2}\lVert q\rVert_{\nu}^{2}+a\langle\zeta^{c}_{\nu},q\rangle_{\nu}-G(\nu),$$

and $(F_{\mathrm{L}})^{\mp}_{\delta}$ for its $\delta$-shifts; by The Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure §equation, viscosity sub- and supersolutions of the lifted equation are those of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution and Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §supersolution for $F_{\mathrm{L}}$ relative to this pair.

Step 1 (growth and exact envelopes). Let $b\in\mathbb{R}$ satisfy $|w(x)|\le b$ for every $x$ (Bounded Real-Valued Function on a Set). By Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral, applied with $m=d$ and $c=w$, the function $\rho\mapsto\int w\,d\rho$ on $\mathcal{P}_{2}(\mathbb{R}^{d})$ is uniformly continuous with absolute value at most $b$; so it is continuous (A Uniformly Continuous Map Between Metric Spaces Is Continuous §continuous), its restriction $W$ is continuous on $\mathcal{D}$ relative to $\mathcal{D}$ (claim 1 of Restriction Stability of Continuity and of the Derivative), and $|W(\mu)|\le b$ for $\mu\in\mathcal{D}$. Since $0\le\mathcal{E}(\mu)$ for $\mu\in\mathcal{D}$ (The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §nonnegative), for every positive $\delta$ and every $\mu\in\mathcal{D}$ we have $W(\mu)\le b+\delta\,\mathcal{E}(\mu)$ and $-b-\delta\,\mathcal{E}(\mu)\le W(\mu)$; thus $W$ has penalty-subordinate growth from above and from below (Penalty-Subordinate Growth of a Function on the Penalty Domain §above, Penalty-Subordinate Growth of a Function on the Penalty Domain §below, with $C=b$). As $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ (The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §lsc), Basic Properties of the Delta-Envelopes on the Wasserstein Space §exact gives, for every positive $\delta$, $W^{-}_{\delta}=W-\delta\mathcal{E}$ and $W^{+}_{\delta}=W+\delta\mathcal{E}$ on $\mathcal{D}$.

Step 2 (the operator along $\nabla w$). Let $E_{w}$, $Dw$ and $\nabla w$ be as in The Almost-Everywhere Gradient of a Lipschitz Semiconvex or Semiconcave Function Is Tangent at Every Absolutely Continuous Measure; these are the objects of the same names in Integrating a Semiconvex Viscosity Subsolution of the Penalty-Drift Equation against a Measure of Finite Relative Fisher Information for $D=\mathbb{R}^{d}$, and by The Almost-Everywhere Gradient of a Lipschitz Semiconvex or Semiconcave Function Is Tangent at Every Absolutely Continuous Measure §borel, $\lVert\nabla w(x)\rVert\le L$ for every $x\in\mathbb{R}^{d}$. For $x\in E_{w}$ one has $\nabla w(x)=Dw(x)$, so $0\le\lVert Dw(x)\rVert\le L$ (the left inequality by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square), and squaring these nonnegative numbers gives $\lVert Dw(x)\rVert^{2}\le L^{2}$ for every $x\in E_{w}$. We claim: for every $\mu\in\mathcal{D}_{\Sigma}$ and $Y\in\mathcal{S}(d)$, $F_{\mathrm{L}}(\mu,W(\mu),\nabla w,Y)\le0$ under the hypotheses of clause 1, and $F_{\mathrm{L}}(\mu,W(\mu),\nabla w,Y)\ge0$ under those of clause 2. Apply Integrating a Semiconvex Viscosity Subsolution of the Penalty-Drift Equation against a Measure of Finite Relative Fisher Information with $D=\mathbb{R}^{d}$ (open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, convex directly from Convex Subset of $\mathbb{R}^n$), $U=V_{c}$, $\lambda=\lambda_{0}$, $\kappa=2a$ (so that its $a=\kappa/2$ is our $a$), $\theta'=\theta$, the running cost $\tilde{g}$, and $A=L^{2}$, $B=0$, $p_{0}=0$ in the gradient bound. Its operator, $\lambda_{0}r+\tfrac{\theta}{2}\lVert p\rVert^{2}+DV_{c}(x)\cdot p-\tfrac{2a}{2}\operatorname{tr}(X)-\tilde{g}(x)$, is the formula of The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space §operator for the data of the statement, so it is $F$, and in both items viscosity sub- and supersolutions are those of Viscosity Subsolution and Supersolution of a Second-Order Equation (The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space §equation). As $D=\mathbb{R}^{d}$, its zero extensions $\bar{w}$ and $\bar{g}$ are $w$ and $\tilde{g}$. Let $\mu\in\mathcal{D}_{\Sigma}$; it has finite relative entropy and finite Fisher information relative to $\gamma_{c}$, so by Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure: Comparison with the Entropy, the Score and the Relative Free Energy of a Quadratic Potential §relative-score, $\mu\in\mathcal{D}^{\Sigma}_{V_{c},a}$ and $\Sigma_{V_{c},a}(\mu)=a\,\zeta^{c}_{\mu}$. The functions $w$, $\tilde{g}$ and $\lVert\nabla w\rVert^{2}$ are Borel and bounded, hence $\mu$-integrable (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures), and by linearity of the integral (Linearity and Monotonicity of the Lebesgue Integral §integrable), Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu and bilinearity of the inner product,

$$\int\Bigl(\lambda_{0}w+\tfrac{\theta}{2}\lVert\nabla w\rVert^{2}-\tilde{g}\Bigr)d\mu+\bigl\langle\Sigma_{V_{c},a}(\mu),\nabla w\bigr\rangle_{\mu}=\lambda_{0}W(\mu)+\tfrac{\theta}{2}\lVert\nabla w\rVert_{\mu}^{2}+a\langle\zeta^{c}_{\mu},\nabla w\rangle_{\mu}-G(\mu)=F_{\mathrm{L}}(\mu,W(\mu),\nabla w,Y).$$

Under the hypotheses of clause 1 this is $\le0$ by Integrating a Semiconvex Viscosity Subsolution of the Penalty-Drift Equation against a Measure of Finite Relative Fisher Information §subsolution; under those of clause 2 it is $\ge0$ by Integrating a Semiconvex Viscosity Subsolution of the Penalty-Drift Equation against a Measure of Finite Relative Fisher Information §supersolution.

Step 3 (witnesses at the touching point). Let $\hat{\mu}\in\mathcal{D}_{\Sigma}$, let $\varphi$ be an intrinsic test function on $\mathcal{D}$ and $\varepsilon>0$, and take $\nu=\hat{\mu}$, $\pi=(\mathrm{id},\mathrm{id})_{\#}\hat{\mu}$, $q=\nabla\varphi(\hat{\mu})$, $Y=H_{\varphi}(\hat{\mu})$. Then $\pi\in\Pi(\hat{\mu},\hat{\mu})$ with $I(\pi)=0<\varepsilon^{2}$ (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 §pushforward with $S=T=\mathrm{id}$); for any $u:\mathcal{D}\to\mathbb{R}$, $|u(\nu)-u(\hat{\mu})|=0<\varepsilon$; the discrepancy of $q$ and $\nabla\varphi(\hat{\mu})$ along $\pi$ is $\lVert q-\nabla\varphi(\hat{\mu})\rVert_{\hat{\mu}}^{2}=0<\varepsilon^{2}$ (The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §diagonal, Elementary Identities in a Real Inner Product Space §vanishing); and $\lVert Y-H_{\varphi}(\hat{\mu})\rVert=0<\varepsilon$ (Difference of Real Matrices, claim 4 of Properties of the Norm of a Symmetric Real Matrix).

Step 4 (clause 1). By Step 1, $W$ has penalty-subordinate growth from above. Let $0<\delta<1$, let $\varphi$ be an intrinsic test function on $\mathcal{D}$, let $\hat{\mu}\in\mathcal{D}$ be a point at which $W^{-}_{\delta}-\varphi$ has a local maximum relative to $\mathcal{D}$, and let $\varepsilon>0$. By Step 1, $W^{-}_{\delta}-\varphi=W-\delta\mathcal{E}-\varphi$ on $\mathcal{D}$, so Penalised Extrema of the Integral of a Lipschitz Semiconvex Function Relative to the Gaussian Free-Energy Pair: Finite Relative Fisher Information and the First-Order Condition §maximum gives $\hat{\mu}\in\mathcal{D}_{\Sigma}$ and $\nabla\varphi(\hat{\mu})+\delta\,\Sigma(\hat{\mu})=\nabla w$ in $L^{2}(\hat{\mu};\mathbb{R}^{d})$. Take the witnesses of Step 3, and let $s=W^{-}_{\delta}(\hat{\mu})\in\mathbb{R}$, so that $|s-W^{-}_{\delta}(\hat{\mu})|=0<\varepsilon$. Since $s+\delta\,\mathcal{E}(\hat{\mu})=W(\hat{\mu})$ by Step 1 and $q+\delta\,\Sigma(\nu)=\nabla\varphi(\hat{\mu})+\delta\,\Sigma(\hat{\mu})=\nabla w$ in $L^{2}(\hat{\mu};\mathbb{R}^{d})$, The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §shifted and Step 2 give

$$(F_{\mathrm{L}})^{-}_{\delta}(\nu,s,q,Y)=F_{\mathrm{L}}\bigl(\hat{\mu},W(\hat{\mu}),\nabla w,Y+\delta H_{\mathcal{E}}(\hat{\mu})\bigr)\le0<\varepsilon .$$

Every requirement of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution is met, so $W$ is a viscosity subsolution of the lifted equation.

Step 5 (clause 2). By Step 1, $W$ has penalty-subordinate growth from below. Let $0<\delta<1$, $\varphi$ an intrinsic test function on $\mathcal{D}$, $\hat{\mu}\in\mathcal{D}$ a point at which $W^{+}_{\delta}-\varphi=W+\delta\mathcal{E}-\varphi$ (Step 1) has a local minimum relative to $\mathcal{D}$, and $\varepsilon>0$. By Penalised Extrema of the Integral of a Lipschitz Semiconvex Function Relative to the Gaussian Free-Energy Pair: Finite Relative Fisher Information and the First-Order Condition §minimum, $\hat{\mu}\in\mathcal{D}_{\Sigma}$ and $\nabla\varphi(\hat{\mu})-\delta\,\Sigma(\hat{\mu})=\nabla w$. Take the witnesses of Step 3, and let $s=W^{+}_{\delta}(\hat{\mu})\in\mathbb{R}$, so that $|s-W^{+}_{\delta}(\hat{\mu})|=0<\varepsilon$ and $s-\delta\,\mathcal{E}(\hat{\mu})=W(\hat{\mu})$, and with $q-\delta\,\Sigma(\nu)=\nabla\varphi(\hat{\mu})-\delta\,\Sigma(\hat{\mu})=\nabla w$ in $L^{2}(\hat{\mu};\mathbb{R}^{d})$, The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §shifted and Step 2 give

$$(F_{\mathrm{L}})^{+}_{\delta}(\nu,s,q,Y)=F_{\mathrm{L}}\bigl(\hat{\mu},W(\hat{\mu}),\nabla w,Y-\delta H_{\mathcal{E}}(\hat{\mu})\bigr)\ge0>-\varepsilon ,$$

and every requirement of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §supersolution is met, so $W$ is a viscosity supersolution of the lifted equation. $\blacksquare$