Proof of Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift and a Score-Paired Wick-Square Cost on a Hilbert Space, 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. Here $W$ denotes a bound of the Wick couplings $w$ (The Wick-Square Corrector and the Score-Paired Wick-Square Cost Relative to a Diagonal Gaussian Measure on a Hilbert Space §couplings), so that Gaussian Dressing of the Score-Paired Wick-Square Cost on a Hilbert Space: Riccati Coefficients, Dressed Variances, the Quadratic Profile, the Constant, the Dressed Pair and the Shifted Operator applies to the data of the theorem.

Notation. Let $P=(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ and $P'=(\mathcal{D}',\mathcal{D}'_{\Sigma},\mathcal{E}',\Sigma')$. By Gaussian Dressing of the Score-Paired Wick-Square Cost on a Hilbert Space: Riccati Coefficients, Dressed Variances, the Quadratic Profile, the Constant, the Dressed Pair and the Shifted Operator §variances, $c'$ is a variance sequence and $\kappa'$ is positive with $c'_{k}\le\kappa'a_{k}$ for every $k$, so A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation may be read with $c'$ in place of $c$, with reference measure $\rho'=\gamma_{c'}$, and $P'$ is defined. By Gaussian Dressing of the Score-Paired Wick-Square Cost on a Hilbert Space: Riccati Coefficients, Dressed Variances, the Quadratic Profile, the Constant, the Dressed Pair and the Shifted Operator §pairs, $\mathcal{P}^{a}_{\rho'}=\mathcal{P}^{a}_{\rho}$, $\mathcal{D}'=\mathcal{D}$, $\mathcal{D}'_{\Sigma}=\mathcal{D}_{\Sigma}$, and, with the real number $C_{0}=\frac{\beta}{2}\sum_{k=1}^{\infty}\log(c'_{k}/c_{k})$ (the series converges absolutely by that clause, hence converges by An Absolutely Convergent Series of Real Numbers Converges §convergence),

$$\mathcal{E}'(\mu)=\mathcal{E}(\mu)+\theta\,\Phi_{0}(\mu)+C_{0}\quad(\mu\in\mathcal{D}),\qquad\Sigma'(\nu)=\Sigma(\nu)+\theta\,\nabla\Phi_{0}(\nu)\quad(\nu\in\mathcal{D}_{\Sigma}).\tag{1}$$

By The Gaussian Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §nonnegative, applied to $P$, and by Gaussian Dressing of the Score-Paired Wick-Square Cost on a Hilbert Space: Riccati Coefficients, Dressed Variances, the Quadratic Profile, the Constant, the Dressed Pair and the Shifted Operator §penalty-pair, $0\le\mathcal{E}(\mu)$ and $0\le\mathcal{E}'(\mu)$ for every $\mu\in\mathcal{D}$.

The reference measure. $P$ is a noise penalty pair on $\mathcal{P}^{a}_{\rho}$ by The Gaussian Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair, and $P'$ is a noise penalty pair on $\mathcal{P}^{a}_{\rho}$, in the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation read with the reference measure $\rho=\gamma_{c}$, by Gaussian Dressing of the Score-Paired Wick-Square Cost on a Hilbert Space: Riccati Coefficients, Dressed Variances, the Quadratic Profile, the Constant, the Dressed Pair and the Shifted Operator §penalty-pair. The metric $W_{a}$ (The Noise Wasserstein Distance §distance), the spaces $L^{2}(\nu;X^{a})$, the noise tangent spaces $T^{a}_{\nu}$ (The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §tangent), noise intrinsic test functions (Noise Intrinsic Test Functions on the Noise Wasserstein Space §test) and their gradients along noise couplings (Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient), including the couplings of vanishing noise cost and strong convergence of Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost and the pairing of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion used in them, the bundle and the $\delta$-shifts (The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts), penalty-subordinate growth (Penalty-Subordinate Growth of a Function on the Domain of a Noise Penalty Pair), the $\delta$-envelopes (The Delta-Envelopes of a Function on the Domain of a Noise Penalty Pair) and the viscosity notions of Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair are defined in terms of the set $\mathcal{P}^{a}_{\rho}=\mathcal{P}^{a}_{\rho'}$ and the pair only, never the reference measure itself, so the notions relative to $P'$, as well as boundedness and uniform continuity of a function on $\mathcal{D}$ relative to $\mathcal{D}$ in the metric space $(\mathcal{P}^{a}_{\rho},W_{a})=(\mathcal{P}^{a}_{\rho'},W_{a})$, are the same whether the setting is read with $\gamma_{c}$ or with $\gamma_{c'}$; hence Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution, read with $c'$ in place of $c$, applies to these notions.

The operators. Let $F$ be the Hamilton-Jacobi operator with Gaussian score drift and the Wick-square cost relative to $\gamma_{c}$ with temperature $\beta$, discount $\lambda_{0}$, control cost $\theta$, Wick couplings $w$ and running cost $g$; by The Hamilton-Jacobi Equation with Gaussian Score Drift and a Score-Paired Wick-Square Cost on a Hilbert Space §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 on the Noise Wasserstein Space Relative to a Noise Penalty Pair and a Profile §subsolution, Viscosity Subsolutions, Supersolutions and Solutions on the Noise Wasserstein Space Relative to a Noise Penalty Pair and a Profile §supersolution or Viscosity Subsolutions, Supersolutions and Solutions on the Noise Wasserstein Space Relative to a Noise Penalty Pair and a Profile §solution. Let $g_{e}:\mathcal{D}\to\mathbb{R}$, $g_{e}(\nu)=g(\nu)+e$, and let $F'$ be the operator of The Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space §operator relative to $\gamma_{c'}$, with temperature $\beta$, discount $\lambda_{0}$, control cost $\theta$ and running cost $g_{e}$ on $\mathcal{D}'=\mathcal{D}$. We write (GS$'$) for the Hamilton-Jacobi equation with Gaussian score drift relative to $\gamma_{c'}$ and the pair $P'$ with these data; by The Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space §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 First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §subsolution, Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §supersolution and Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §solution.

By Gaussian Dressing of the Score-Paired Wick-Square Cost on a Hilbert Space: Riccati Coefficients, Dressed Variances, the Quadratic Profile, the Constant, the Dressed Pair and the Shifted Operator §operator, $F^{\Phi_{0}}(\nu,r,q)=F'(\nu,r,q)$ for all $(\nu,q)\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$ and $r\in\mathbb{R}$, where $F^{\Phi_{0}}$ is the operator $F$ shifted by $\Phi_{0}$. So $F^{\Phi_{0}}$ and $F'$ are the same first-order equation operator over $\mathcal{D}_{\Sigma}$, and their $\delta$-shifts relative to $P$, which The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts §shifted determines from the operator and the pair, coincide for every positive $\delta$. The conditions of Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §subsolution and Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §supersolution involve the operator only through these $\delta$-shifts. Hence:

(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. By Gaussian Dressing of the Score-Paired Wick-Square Cost on a Hilbert Space: Riccati Coefficients, Dressed Variances, the Quadratic Profile, the Constant, the Dressed Pair and the Shifted Operator §riccati, the Riccati coefficients $b$ form an admissible sequence, and $\Phi_{0}=\Phi_{b}$ by Gaussian Dressing of the Score-Paired Wick-Square Cost on a Hilbert Space: Riccati Coefficients, Dressed Variances, the Quadratic Profile, the Constant, the Dressed Pair and the Shifted Operator §profile. By Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions §scaling, applied with $b$ and $\theta$, the sequence $\theta b$ is admissible, $\Phi_{\theta b}(\mu)=\theta\,\Phi_{0}(\mu)$ and $V_{\theta b}(\mu)=\theta\,V_{b}(\mu)$ for $\mu\in\mathcal{P}^{a}_{\rho}$. Put $\Psi=\Phi_{\theta b}$. Since $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ (The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §pair), Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions §test, applied with $\theta b$ and $Q=\mathcal{D}$, shows that $\Psi$ is a noise intrinsic test function on $\mathcal{D}$ with $\nabla\Psi(\nu)=V_{\theta b}(\nu)=\theta\,V_{b}(\nu)=\theta\,\nabla\Phi_{0}(\nu)$ for $\nu\in\mathcal{D}$, the last equality by Gaussian Dressing of the Score-Paired Wick-Square Cost on a Hilbert Space: Riccati Coefficients, Dressed Variances, the Quadratic Profile, the Constant, the Dressed Pair and the Shifted Operator §profile. So (1) reads

$$\mathcal{E}'(\mu)=\mathcal{E}(\mu)+\Psi(\mu)+C_{0}\quad(\mu\in\mathcal{D}),\qquad\Sigma'(\nu)=\Sigma(\nu)+\nabla\Psi(\nu)\quad(\nu\in\mathcal{D}_{\Sigma}),$$

and $P$, $P'$ are noise penalty pairs on $\mathcal{P}^{a}_{\rho}$ (The Gaussian Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair and Gaussian Dressing of the Score-Paired Wick-Square Cost on a Hilbert Space: Riccati Coefficients, Dressed Variances, the Quadratic Profile, the Constant, the Dressed Pair and the Shifted Operator §penalty-pair) with the same penalty domain and score domain; these are the hypotheses of Changing the Penalty of a Noise Penalty Pair by a Noise Intrinsic Test Function and a Constant Does Not Change the Viscosity Notion §pair for $P$, $P'$, $\Psi$ and $C_{0}$.

Step 2: bounded functions have subordinate growth. Let $v:\mathcal{D}\to\mathbb{R}$ and $B\in\mathbb{R}$ with $v(\mu)\le B$ for every $\mu\in\mathcal{D}$. For every positive $\delta$, $v(\mu)\le B\le B+\delta\,\mathcal{E}(\mu)$ and $v(\mu)\le B\le B+\delta\,\mathcal{E}'(\mu)$ for every $\mu\in\mathcal{D}$, the penalties being nonnegative; so $v$ has penalty-subordinate growth from above relative to $P$ and relative to $P'$, with the constant $B$. Likewise, let $B'\in\mathbb{R}$ with $B'\le v(\mu)$ for every $\mu\in\mathcal{D}$, and take $-B'$ as the constant of Penalty-Subordinate Growth of a Function on the Domain of a Noise Penalty Pair §below (written $C$ there); for every positive $\delta$ and every $\mu\in\mathcal{D}$, the penalties being nonnegative, $-(-B')-\delta\,\mathcal{E}(\mu)=B'-\delta\,\mathcal{E}(\mu)\le B'\le v(\mu)$ and $-(-B')-\delta\,\mathcal{E}'(\mu)=B'-\delta\,\mathcal{E}'(\mu)\le B'\le v(\mu)$, so $v$ has penalty-subordinate growth from below relative to $P$ and to $P'$, with the constant $-B'$.

Step 3: one-sided transfer. Let $U:\mathcal{D}\to\mathbb{R}$ and $Y=U-\Phi_{0}$.

(i) Suppose $Y(\mu)\le B$ for every $\mu\in\mathcal{D}$, for some $B\in\mathbb{R}$. By Step 2, $Y$ 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 $\Phi_{0}$; $Y$ is a viscosity subsolution of $F^{\Phi_{0}}$ relative to $P$ (by Viscosity Subsolutions, Supersolutions and Solutions on the Noise Wasserstein Space Relative to a Noise Penalty Pair and a Profile §subsolution); $Y$ is a viscosity subsolution of $F'$ relative to $P$ (by (C)); $Y$ is a viscosity subsolution of $F'$ relative to $P'$ (by Changing the Penalty of a Noise Penalty Pair by a Noise Intrinsic Test Function and a Constant Does Not Change the Viscosity Notion §pair and Step 1); $Y$ is a viscosity subsolution of (GS$'$) (by the notation paragraph and the paragraph on the reference measure).

(ii) Suppose $B'\le Y(\mu)$ for every $\mu\in\mathcal{D}$, for some $B'\in\mathbb{R}$. By Step 2, $Y$ has penalty-subordinate growth from below relative to $P$ and to $P'$, and the same chain, with Viscosity Subsolutions, Supersolutions and Solutions on the Noise Wasserstein Space Relative to a Noise Penalty Pair and a Profile §supersolution and the supersolution half of Changing the Penalty of a Noise Penalty Pair by a Noise Intrinsic Test Function and a Constant Does Not Change the Viscosity Notion §pair, shows that $U$ is a viscosity supersolution relative to the profile $\Phi_{0}$ if and only if $Y$ is a viscosity supersolution of (GS$'$).

Clause 4. Let $U-\Phi_{0}$ be bounded, with a bound $K$, so that $-K\le U(\mu)-\Phi_{0}(\mu)\le K$ on $\mathcal{D}$. By Step 2, $U-\Phi_{0}$ 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 on the Noise Wasserstein Space Relative to a Noise Penalty Pair and a Profile §solution and Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §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 3 with $B=K$ and $B'=-K$, if and only if $U-\Phi_{0}$ is a viscosity subsolution and a viscosity supersolution of (GS$'$); by the second, if and only if $U-\Phi_{0}$ is a viscosity solution of (GS$'$).

The data of (GS$'$). We check the hypotheses of Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution, read with $c'$ in place of $c$, for $\beta$, the constant $\kappa'$, the pair $P'$, $\lambda_{0}$ and $\theta$ (with $0<\lambda_{0}$ and $0<\theta\le1$) and the running cost $g_{e}$ on $\mathcal{D}'=\mathcal{D}$. First, $0\le C+|e|$ and $|g_{e}(\mu)|\le|g(\mu)|+|e|\le C+|e|$ for every $\mu\in\mathcal{D}$; 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 corollary. Given a positive $\varepsilon$, let $\delta$ be the positive number supplied for $g$ and $\varepsilon$ by Uniformly Continuous Map Between Metric Spaces; if $\mu,\mu'\in\mathcal{D}$ and $W_{a}(\mu,\mu')<\delta$, then $|g_{e}(\mu)-g_{e}(\mu')|=|g(\mu)-g(\mu')|<\varepsilon$, so $g_{e}$ is uniformly continuous on $\mathcal{D}$; by the paragraph on the reference measure, the metric space $(\mathcal{P}^{a}_{\rho'},W_{a})$ of that corollary is $(\mathcal{P}^{a}_{\rho},W_{a})$. The viscosity notions of that corollary are those of (GS$'$).

Clause 1. By Step 3(i) with $B=B_{1}$, $u-\Phi_{0}$ is a viscosity subsolution of (GS$'$), and by Step 3(ii) with $B'=B_{2}$, $v-\Phi_{0}$ is a viscosity supersolution of (GS$'$). Since $u-\Phi_{0}\le B_{1}$ and $B_{2}\le v-\Phi_{0}$ on $\mathcal{D}$, Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space: 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 Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §existence, with the constant $C+|e|$, there is a viscosity solution $u_{\ast}:\mathcal{D}\to\mathbb{R}$ of (GS$'$) with $-\lambda_{0}^{-1}(C+|e|)\le u_{\ast}(\mu)\le\lambda_{0}^{-1}(C+|e|)$ for every $\mu\in\mathcal{D}$. Put $U(\mu)=u_{\ast}(\mu)+\Phi_{0}(\mu)$ for $\mu\in\mathcal{D}$. Then $U-\Phi_{0}=u_{\ast}$ is bounded, with bound $\lambda_{0}^{-1}(C+|e|)\ge0$, and is a viscosity solution of (GS$'$), so $U$ is a viscosity solution relative to the profile $\Phi_{0}$ by clause 4; the asserted bounds are those of $u_{\ast}$.

Clause 3. By clause 4, $U-\Phi_{0}$ and $U'-\Phi_{0}$ are viscosity solutions of (GS$'$), each bounded. By Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §uniqueness they are equal, so $U(\mu)=U'(\mu)$ for every $\mu\in\mathcal{D}$.