Proof of Well-Posedness of the Lifted Hamilton-Jacobi Equation with a Wick-Square Cost Relative to a Diagonal Gaussian Measure, by Gaussian Dressing

Each result cited is universally quantified over the data in its own statement. Elementary ordered-field arithmetic, order and the properties of the absolute value are used without further comment, as provided by The Real Numbers: Standing Notation and Background §background.

Notation. Let $P=(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian free-energy pair with variances $c$ and temperature $a$, and $P'=(\mathcal{D}',\mathcal{D}'_{\Sigma},\mathcal{E}',\Sigma')$ the one with variances $c'$ and temperature $a$; $c'$ is a variance vector by Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator §variances, and $\mathcal{D}'=\mathcal{D}$, $\mathcal{D}'_{\Sigma}=\mathcal{D}_{\Sigma}$ by Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator §pairs. Both are penalty pairs 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 Wasserstein-closed by The Gaussian Free-Energy Pair is Wasserstein-Closed §w2-closed. Let $F$ be the lifted Wick-square Hamilton-Jacobi operator with variances $c$, temperature $a$, discount $\lambda_{0}$, control cost $\theta$, couplings $\beta$ and running cost $g$, so that, by The Lifted Hamilton-Jacobi Equation with a Wick-Square Cost Relative to a Diagonal Gaussian Measure §equation, a viscosity subsolution, supersolution or solution relative to the profile $\Phi_{0}$ is one of $F$ relative to $P$ and the profile $\Phi_{0}$ in the sense of Viscosity Subsolutions, Supersolutions and Solutions Relative to a Penalty Pair and a Profile §subsolution, Viscosity Subsolutions, Supersolutions and Solutions Relative to a Penalty Pair and a Profile §supersolution or Viscosity Subsolutions, Supersolutions and Solutions Relative to a Penalty Pair and a Profile §solution. Let $g_{e}:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$, $g_{e}(\nu)=g(\nu)+e$, and let $F'$ be the lifted Ornstein-Uhlenbeck Hamilton-Jacobi operator of $P'$ with discount $\lambda_{0}$, control cost $\theta$ and running cost $g_{e}$, as in The Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure §operator. We write (OU$'$) for the lifted Ornstein-Uhlenbeck Hamilton-Jacobi equation with variances $c'$, temperature $a$, discount $\lambda_{0}$, control cost $\theta$ and running cost $g_{e}$; by The Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure §equation, its viscosity subsolutions, supersolutions and solutions are those of $F'$ relative to $P'$ in the sense of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution, Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §supersolution and Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §solution.

By Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator §operator, $F^{\Phi_{0}}(\nu,r,q,Y)=F'(\nu,r,q,Y)$ for all $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$ and $Y\in\mathcal{S}(d)$, where $F^{\Phi_{0}}$ is the operator of Viscosity Subsolutions, Supersolutions and Solutions Relative to a Penalty Pair and a Profile §shifted. Thus $F^{\Phi_{0}}$ and $F'$ are equal as functions on $\mathcal{V}(\mathcal{D}_{\Sigma})\times\mathbb{R}\times\mathcal{S}(d)$, that is, they are the same second-order equation operator over $\mathcal{D}_{\Sigma}$, and so their $\delta$-shifts relative to $P$, which by The Intrinsic Calculus on the Wasserstein Space: Standing Notation §operators are determined by the operator and the pair, coincide for every $\delta\in(0,1)$. For a function $v:\mathcal{D}\to\mathbb{R}$ with penalty-subordinate growth from above (respectively below) relative to $P$, the conditions of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution (respectively Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §supersolution) for $F^{\Phi_{0}}$ and for $F'$ relative to $P$ involve the operators only through these $\delta$-shifts, so they are literally the same. Hence we have the following statement (C): for $v:\mathcal{D}\to\mathbb{R}$ with penalty-subordinate growth from above (respectively below) relative to $P$, $v$ is a viscosity subsolution (respectively supersolution) of $F^{\Phi_{0}}$ relative to $P$ if and only if it is one of $F'$ relative to $P$.

Step 1: the change of penalty. Let $B$ be the diagonal matrix with diagonal entries $b_{1},\dots,b_{d}$, put $\kappa_{0}=a\,(Z_{c'}-Z_{c})$ and $f(z)=\theta\,u_{0}(z)+\kappa_{0}=\tfrac12\,z\cdot((\theta B)z)+\kappa_{0}$ for $z\in\mathbb{R}^{d}$, where $u_{0}$ is the function of Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator §profile. By Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic, applied with the symmetric matrix $\theta B$, linear coefficient $0$ and constant term $\kappa_{0}$, $f$ is of class $C^{2}$ on $\mathbb{R}^{d}$ with $Df(z)=\theta Bz=\theta\,Du_{0}(z)$ and Hessian matrix $\theta B$, so that $|\partial_{j}\partial_{i}f(z)|\le\theta\sum_{k=1}^{d}|b_{k}|$ for all $z$ and $i,j$. By Integrals of Functions with a Bounded Hessian are Intrinsic Test Functions on the Wasserstein Space §test, $\Psi=\varphi_{f}$, $\Psi(\mu)=\int f\,d\mu$, is an intrinsic test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$ with $\nabla\Psi(\mu)$ the class $\theta\,Du_{0}$; and since $u_{0}$ is $\mu$-integrable by Integrals of Functions with a Bounded Hessian are Intrinsic Test Functions on the Wasserstein Space §integrable and $\mu$ is a probability measure, Linearity and Monotonicity of the Lebesgue Integral §integrable gives $\Psi(\mu)=\theta\,\Phi_{0}(\mu)+\kappa_{0}$. By Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §restriction, $\Psi$ is an intrinsic test function on $\mathcal{D}$ with the same gradients, and by Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator §profile the class $\theta\,Du_{0}$ in $L^{2}(\nu;\mathbb{R}^{d})$ is $\theta\,\nabla\Phi_{0}(\nu)$. Therefore Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator §pairs reads

$$\mathcal{E}'(\mu)=\mathcal{E}(\mu)+\Psi(\mu)\quad(\mu\in\mathcal{D}),\qquad\Sigma'(\nu)=\Sigma(\nu)+\nabla\Psi(\nu)\quad(\nu\in\mathcal{D}_{\Sigma}),$$

and $P'$ has the same penalty domain and score domain as $P$; these are the hypotheses of Shifting the Penalised Viscosity Notion by an Intrinsic Test Function: Composition of Shifts, the Shift Identity, Invariance under a Change of Profile, and a Change of the Penalty §pair for $P$, $P'$ and $\Psi$.

Step 2: one-sided transfer. Let $U:\mathcal{D}\to\mathbb{R}$ and $W=U-\Phi_{0}$.

(i) Suppose $W(\mu)\le k$ for every $\mu\in\mathcal{D}$, for some $k\in\mathbb{R}$. By The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Closed Penalty Pair §growth, applied to the Wasserstein-closed pairs $P$ and $P'$, $W$ has penalty-subordinate growth from above relative to $P$ and relative to $P'$. Then the following are equivalent: $U$ is a viscosity subsolution relative to the profile $\Phi_{0}$; $W$ is a viscosity subsolution of $F^{\Phi_{0}}$ relative to $P$ (by Viscosity Subsolutions, Supersolutions and Solutions Relative to a Penalty Pair and a Profile §subsolution); $W$ is a viscosity subsolution of $F'$ relative to $P$ (by (C)); $W$ is a viscosity subsolution of $F'$ relative to $P'$ (by Shifting the Penalised Viscosity Notion by an Intrinsic Test Function: Composition of Shifts, the Shift Identity, Invariance under a Change of Profile, and a Change of the Penalty §pair and Step 1); $W$ is a viscosity subsolution of (OU$'$).

(ii) Suppose $k'\le W(\mu)$ for every $\mu\in\mathcal{D}$, for some $k'\in\mathbb{R}$. By The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Closed Penalty Pair §growth, $W$ has penalty-subordinate growth from below relative to $P$ and to $P'$, and the same chain, with Viscosity Subsolutions, Supersolutions and Solutions Relative to a Penalty Pair and a Profile §supersolution and the supersolution half of Shifting the Penalised Viscosity Notion by an Intrinsic Test Function: Composition of Shifts, the Shift Identity, Invariance under a Change of Profile, and a Change of the Penalty §pair, shows that $U$ is a viscosity supersolution relative to the profile $\Phi_{0}$ if and only if $W$ is a viscosity supersolution of (OU$'$).

Clause 4. Let $U-\Phi_{0}$ be bounded (Bounded Real-Valued Function on a Set), say $|U(\mu)-\Phi_{0}(\mu)|\le K$ on $\mathcal{D}$. By The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Closed Penalty Pair §growth, $U-\Phi_{0}$ has penalty-subordinate growth from above and from below relative to $P$ and to $P'$, so the solution notions of Viscosity Subsolutions, Supersolutions and Solutions Relative to a Penalty Pair and a Profile §solution and Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §solution apply. By the first, $U$ is a viscosity solution relative to the profile $\Phi_{0}$ if and only if it is a subsolution and a supersolution relative to that profile; by Step 2 with $k=K$ and $k'=-K$, if and only if $U-\Phi_{0}$ is a viscosity subsolution and a viscosity supersolution of (OU$'$); by the second, if and only if $U-\Phi_{0}$ is a viscosity solution of (OU$'$).

The data of (OU$'$). We check the hypotheses of Well-Posedness of the Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution for the variance vector $c'$, the temperature $a$, the pair $P'$, $\lambda_{0}$, $\theta$ (with $0<\lambda_{0}$ and $0<\theta\le1$) and the running cost $g_{e}$. First, $0\le C$: the Dirac measure $\mu_{\circ}$ at $0_{\mathbb{R}^{d}}$ lies in $\mathcal{P}_{2}(\mathbb{R}^{d})$ by Dirac Measures on Euclidean Space: Probability Measure, Integrals, Push-Forwards and the Coupling of Two Dirac Measures §measure (with $n=d$), so $0\le|g(\mu_{\circ})|\le C$. Hence $0\le C+|e|$, as $0\le|e|$. For every $\mu$, $|g_{e}(\mu)|\le|g(\mu)|+|e|\le C+|e|$, so $g_{e}$ is bounded in the sense of Bounded Real-Valued Function on a Set and $C+|e|$ may serve as the constant $C$ of that theorem. Given a positive $\varepsilon$, let $\delta$ be the positive number supplied for $g$ and $\varepsilon$ by Uniformly Continuous Map Between Metric Spaces; if $W_{2}(\mu,\mu')<\delta$ then $|g_{e}(\mu)-g_{e}(\mu')|=|g(\mu)-g(\mu')|<\varepsilon$, so $g_{e}$ is uniformly continuous. Viscosity notions in that theorem are those of (OU$'$).

Clause 1. By Step 2(i) with the bound $k$, $u-\Phi_{0}$ is a viscosity subsolution of (OU$'$), and by Step 2(ii) with the bound $k'$, $v-\Phi_{0}$ is a viscosity supersolution of (OU$'$). Since $u-\Phi_{0}\le k$ and $k'\le v-\Phi_{0}$ on $\mathcal{D}$, Well-Posedness of the Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §comparison gives $u(\mu)-\Phi_{0}(\mu)\le v(\mu)-\Phi_{0}(\mu)$, hence $u(\mu)\le v(\mu)$, for every $\mu\in\mathcal{D}$.

Clause 2. By Well-Posedness of the Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §existence, with the constant $C+|e|$ (nonnegative, as shown in the data of (OU$'$), so that also $0\le\lambda_{0}^{-1}(C+|e|)$, as $0<\lambda_{0}^{-1}$), there is a viscosity solution $w:\mathcal{D}\to\mathbb{R}$ of (OU$'$) with $-\lambda_{0}^{-1}(C+|e|)\le w(\mu)\le\lambda_{0}^{-1}(C+|e|)$ on $\mathcal{D}$. Put $U(\mu)=w(\mu)+\Phi_{0}(\mu)$ for $\mu\in\mathcal{D}$. Then $U-\Phi_{0}=w$ is bounded and a viscosity solution of (OU$'$), so $U$ is a viscosity solution relative to the profile $\Phi_{0}$ by clause 4, and the asserted bounds are those of $w$.

Clause 3. By clause 4, $U-\Phi_{0}$ and $U'-\Phi_{0}$ are viscosity solutions of (OU$'$), each bounded. By Well-Posedness of the Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §uniqueness they are equal, so $U(\mu)=U'(\mu)$ for every $\mu\in\mathcal{D}$.

Clause 5. Let $g$ be constant with value $g_{0}$, so $g_{e}$ is constant with value $g_{0}+e$, and put $\kappa=\lambda_{0}^{-1}(g_{0}+e)$, so that $\lambda_{0}\kappa=g_{0}+e=g_{e}(\nu)$ for every $\nu\in\mathcal{D}_{\Sigma}$. By The Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure §operator, $F'$ is the Hamilton-Jacobi operator with common noise and penalty drift of $P'$ with discount $\lambda_{0}$, common-noise matrix $0_{1\times d}$ (so $p=1$), control cost $\theta$ and running cost $g_{e}$. The pair $P'$ is Wasserstein-closed (above) and has regular penalised maxima by The Gaussian Free-Energy Pair is Uniformly Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima §regular. So Constant Viscosity Subsolutions and Supersolutions of the Hamilton-Jacobi Equation with Common Noise and Penalty Drift, with $\Gamma=0_{1\times d}$ and this $\kappa$, shows that the constant function $u_{\kappa}$ with value $\kappa$ has penalty-subordinate growth from above and from below relative to $P'$ (Constant Viscosity Subsolutions and Supersolutions of the Hamilton-Jacobi Equation with Common Noise and Penalty Drift §growth) and is a viscosity subsolution and a viscosity supersolution of $F'$ relative to $P'$ (Constant Viscosity Subsolutions and Supersolutions of the Hamilton-Jacobi Equation with Common Noise and Penalty Drift §subsolution and Constant Viscosity Subsolutions and Supersolutions of the Hamilton-Jacobi Equation with Common Noise and Penalty Drift §supersolution, using $\lambda_{0}\kappa\le g_{e}(\nu)$ and $g_{e}(\nu)\le\lambda_{0}\kappa$). By Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §solution, $u_{\kappa}$ is a viscosity solution of (OU$'$). Since $U_{0}-\Phi_{0}=u_{\kappa}$ is bounded (by $|\kappa|$), clause 4 shows that $U_{0}$ is a viscosity solution relative to the profile $\Phi_{0}$. Finally, if $U$ is a viscosity solution relative to the profile $\Phi_{0}$ with $U-\Phi_{0}$ bounded, clause 3 applied to $U$ and $U_{0}$ gives $U=U_{0}$.