Proof 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

Each result cited is universally quantified over the data in its own statement.

Throughout, $0_{1\times d}$ is the zero real $1\times d$ matrix, $p=1$ is its number of rows, and $F$ is the lifted Ornstein-Uhlenbeck Hamilton-Jacobi operator with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$. By its definition, $F$ is the Hamilton-Jacobi operator with common noise and penalty drift of the Gaussian free-energy pair with discount $\lambda_{0}$, common-noise matrix $0_{1\times d}$, control cost $\theta$ and running cost $g$; by The Discounted Hamilton-Jacobi Equation with Common Noise and a Penalty Drift on the Wasserstein Space §operator it is a second-order equation operator over $\mathcal{D}_{\Sigma}$, and its $\delta$-shifts relative to the pair are those of The Intrinsic Calculus on the Wasserstein Space: Standing Notation §operators. The data $\lambda_{0},\theta$ are positive with $\theta\le1$, so all the results below on that operator apply with $\Gamma=0_{1\times d}$. By the definition of the equation, a viscosity subsolution, supersolution or solution of the lifted equation is precisely a viscosity subsolution, supersolution or solution of $F$ relative to the Gaussian free-energy pair 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, which is the notion used in A Comparison Principle for First-Order Equations on the Wasserstein Space Relative to a Wasserstein-Closed Penalty Pair and in Perron's Method on the Wasserstein Space: Existence of a Viscosity Solution Between a Subsolution and a Supersolution. The pair is a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §pair.

Step 0 (properties of the pair). By The Gaussian Free-Energy Pair is Wasserstein-Closed §w2-closed the pair is Wasserstein-closed. By The Gaussian Free-Energy Pair is Uniformly Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima §convex it is displacement convex, by The Gaussian Free-Energy Pair is Uniformly Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima §closed it has closed score along couplings, and by The Gaussian Free-Energy Pair is Uniformly Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima §regular it has regular penalised maxima. By The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §maps its penalty domain $\mathcal{D}$ has the map property, and by The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §lsc the penalty $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$.

Step 1 (properties of the operator). First, $F$ does not depend on its matrix argument: by the displayed value of the operator, for $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$ and $Y\in\mathcal{S}(d)$ one has $F(\nu,r,q,Y)=\lambda_{0}r+\frac{\theta}{2}\lVert q\rVert_{\nu}^{2}+a\langle\zeta^{c}_{\nu},q\rangle_{\nu}-g(\nu)$, an expression in which $Y$ does not occur, so $F(\nu,r,q,Y)=F(\nu,r,q,Y')$ for all $Y,Y'\in\mathcal{S}(d)$.

Next we verify the hypotheses of The Hamilton-Jacobi Operator with Common Noise and Penalty Drift Satisfies the Hypotheses of the Comparison Principle for a Displacement Convex Pair with $p=1$ and $\Gamma=0_{1\times d}$. Convexity holds by Step 0, and semicontinuity holds by Step 0. For growth, every entry of the $d\times d$ matrix $0_{1\times d}^{\top}0_{1\times d}$ is a product of two zero entries, so this matrix is the zero matrix, and hence for every $\mu\in\mathcal{D}$ the product $0_{1\times d}^{\top}0_{1\times d}H_{\mathcal{E}}(\mu)$ is the zero matrix, whose trace is $0$; this is elementary arithmetic carried by The Real Numbers: Standing Notation and Background §background, as in the computation of the operator. Put

$$C_{1}=\frac{4c_{\max}}{a}+2\sum_{i=1}^{d}c_{i},$$

a positive real number because $a>0$ and every $c_{i}$ is positive by The Diagonal Gaussian Density on Euclidean Space and Its Notation §variances. For $\mu\in\mathcal{D}$ one has $0\le\mathcal{E}(\mu)$ by The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §nonnegative, so $|\mathcal{E}(\mu)|=\mathcal{E}(\mu)$, and by The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §moment and elementary order arithmetic,

$$M_{2}(\mu)\le\frac{4c_{\max}}{a}\,\mathcal{E}(\mu)+2\sum_{i=1}^{d}c_{i}\le C_{1}\,\mathcal{E}(\mu)+C_{1}=C_{1}\bigl(1+|\mathcal{E}(\mu)|\bigr),$$

while the trace term has absolute value $|0|=0\le C_{1}(1+|\mathcal{E}(\mu)|)$, since $C_{1}>0$ and $|\mathcal{E}(\mu)|\ge0$. For Hessian continuity, the map $\mu\mapsto\mathrm{tr}(0_{1\times d}^{\top}0_{1\times d}H_{\mathcal{E}}(\mu))$ is the constant map $0$ on $\mathcal{D}$, and the restriction of a constant map to any subset is continuous, since the distance between any two of its values is $0$. The running-cost hypothesis is the assumption that $g$ is bounded and uniformly continuous. Hence, by The Hamilton-Jacobi Operator with Common Noise and Penalty Drift Satisfies the Hypotheses of the Comparison Principle for a Displacement Convex Pair §conclusion, $F$ is locally strictly proper and satisfies the shift-coercivity condition, the shift-semicontinuity condition and the second-order structure condition at uniquely mapped pairs. Further, $F$ has momentum-continuous shifts relative to the pair by The Hamilton-Jacobi Operator with Common Noise and Penalty Drift Has Momentum-Continuous Shifts §momentum, and $F$ is degenerate elliptic in the sense of Degenerate Elliptic Second-Order Equation Operators on the Wasserstein Space §elliptic by The Hamilton-Jacobi Operator with Common Noise and Penalty Drift is Degenerate Elliptic.

Step 2 (comparison, clause 1). Let $u$, $v$, $b$, $b'$ be as in clause 1. By Step 0 the pair is a Wasserstein-closed penalty pair with closed score along couplings whose penalty domain has the map property, and by Step 1 the operator $F$ is a second-order equation operator over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts relative to the pair, that does not depend on its matrix argument, is locally strictly proper, satisfies the shift-coercivity, shift-semicontinuity and second-order structure conditions, and has momentum-continuous shifts. Moreover $u\le b$ and $b'\le v$ on $\mathcal{D}$, $u$ is a viscosity subsolution and $v$ a viscosity supersolution of $F$ relative to the pair. Hence A Comparison Principle for First-Order Equations on the Wasserstein Space Relative to a Wasserstein-Closed Penalty Pair §comparison gives $u(\mu)\le v(\mu)$ for every $\mu\in\mathcal{D}$.

Step 3 (existence, clause 2). Put $\kappa_{-}=-\lambda_{0}^{-1}C$ and $\kappa_{+}=\lambda_{0}^{-1}C$, and let $u_{\kappa_{-}},u_{\kappa_{+}}:\mathcal{D}\to\mathbb{R}$ be the constant functions with values $\kappa_{-}$ and $\kappa_{+}$. Then $\lambda_{0}\kappa_{-}=-C$ and $\lambda_{0}\kappa_{+}=C$ by field arithmetic. For $\nu\in\mathcal{D}_{\Sigma}\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ one has $|g(\nu)|\le C$, hence $-C\le g(\nu)\le C$ by the elementary properties of the absolute value; that is, $\lambda_{0}\kappa_{-}\le g(\nu)$ and $g(\nu)\le\lambda_{0}\kappa_{+}$. The pair is Wasserstein-closed with regular penalised maxima by Step 0, and $F$ is the Hamilton-Jacobi operator with common noise and penalty drift with $\Gamma=0_{1\times d}$; so by Constant Viscosity Subsolutions and Supersolutions of the Hamilton-Jacobi Equation with Common Noise and Penalty Drift §subsolution the function $u_{\kappa_{-}}$ is a viscosity subsolution of $F$ relative to the pair, by Constant Viscosity Subsolutions and Supersolutions of the Hamilton-Jacobi Equation with Common Noise and Penalty Drift §supersolution the function $u_{\kappa_{+}}$ is a viscosity supersolution of $F$ relative to the pair, and by Constant Viscosity Subsolutions and Supersolutions of the Hamilton-Jacobi Equation with Common Noise and Penalty Drift §growth both have penalty-subordinate growth from above and from below. For every $\nu\in\mathcal{D}$ we have $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, so $0\le|g(\nu)|\le C$ and thus $0\le C$; as $\lambda_{0}^{-1}$ is positive, $0\le\lambda_{0}^{-1}C$ and hence $u_{\kappa_{-}}(\nu)=-\lambda_{0}^{-1}C\le\lambda_{0}^{-1}C=u_{\kappa_{+}}(\nu)$, by elementary order arithmetic.

We now apply Perron's Method on the Wasserstein Space: Existence of a Viscosity Solution Between a Subsolution and a Supersolution with the Gaussian free-energy pair, which is Wasserstein-closed with regular penalised maxima and has penalty domain with the map property by Step 0, with the operator $F$, which is degenerate elliptic by Step 1, and with the subsolution $u_{\kappa_{-}}$ (having growth from below) and the supersolution $u_{\kappa_{+}}$ (having growth from above) in the roles of the functions written $f$ and $g$ there (the latter is not the running cost $g$). Let $u:\mathcal{D}\to\mathbb{R}$ be the supremum function defined there. By Perron's Method on the Wasserstein Space: Existence of a Viscosity Solution Between a Subsolution and a Supersolution §solution, $u$ is a viscosity solution of $F$ relative to the pair, hence a viscosity solution of the lifted Ornstein-Uhlenbeck Hamilton-Jacobi equation by The Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure §equation. By Perron's Method on the Wasserstein Space: Existence of a Viscosity Solution Between a Subsolution and a Supersolution §bounds, for every $\mu\in\mathcal{D}$,

$$-\lambda_{0}^{-1}C=u_{\kappa_{-}}(\mu)\le u(\mu)\le u_{\kappa_{+}}(\mu)=\lambda_{0}^{-1}C,$$

which is clause 2.

Step 4 (uniqueness, clause 3). Let $u,u'$ be bounded viscosity solutions. By Bounded Real-Valued Function on a Set there are real numbers $M,M'$ with $|u(\mu)|\le M$ and $|u'(\mu)|\le M'$ for every $\mu\in\mathcal{D}$, hence $-M\le u(\mu)\le M$ and $-M'\le u'(\mu)\le M'$ by the elementary properties of the absolute value. By Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §solution each of $u,u'$ is both a viscosity subsolution and a viscosity supersolution of $F$ relative to the pair, that is, of the lifted equation. Applying clause 1 (Step 2) with the subsolution $u$, the supersolution $u'$, $b=M$ and $b'=-M'$ gives $u(\mu)\le u'(\mu)$ for every $\mu\in\mathcal{D}$; applying it with the subsolution $u'$, the supersolution $u$, $b=M'$ and $b'=-M$ gives $u'(\mu)\le u(\mu)$ for every $\mu\in\mathcal{D}$. By antisymmetry of the order of $\mathbb{R}$, carried by The Real Numbers: Standing Notation and Background §background, $u(\mu)=u'(\mu)$ for every $\mu\in\mathcal{D}$. $\blacksquare$