Proof of Well-Posedness of the Hamilton-Jacobi Equation with Gibbs Score Drift for a Semiconvex Cylindrical Potential: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution

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

Real order and arithmetic, including the properties of absolute values, are used through The Real Numbers: Standing Notation and Background §background, without further mention.

We apply Well-Posedness of the Hamilton-Jacobi Equation with Penalty Drift for a Lambda-Displacement Convex Noise Penalty Pair, Lambda Any Real Number: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution to the Gibbs entropy pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ with potential $V$ and temperature $\beta$. Its settings are in force: The Real Numbers: Standing Notation and Background and Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation are in force by A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §background, with the reference measure $\rho=\gamma_{c}$ by A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §gaussian, and First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation is layered on Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation and only fixes notation. That setting reserves the letter $c$ for real numbers unless a result says otherwise (First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §operators); the statement of Well-Posedness of the Hamilton-Jacobi Equation with Penalty Drift for a Lambda-Displacement Convex Noise Penalty Pair, Lambda Any Real Number: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution uses that letter only as the subscript of $\lambda_{c}$, which is a label, so no conflict arises with the variance sequence $c$ of the present setting, and the real number quantified as $c$ in the noise-closedness definition is a bound variable, as noted in The Gibbs Entropy Pair is Noise-Closed, and Its Penalty Bounds the Squared Noise Wasserstein Distance to the Reference Measure. The theorem names a growth constant $K$, which would clash with the semiconvexity constant $K$ of the present statement; we write $K'$ for the theorem's constant. The quadruple $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is a noise penalty pair on $\mathcal{P}^{a}_{\rho}$ by The Gibbs Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair, and the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §space is that of The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §metric, which is the one used in the statement of the corollary.

The hypothesis (The pair). We verify Well-Posedness of the Hamilton-Jacobi Equation with Penalty Drift for a Lambda-Displacement Convex Noise Penalty Pair, Lambda Any Real Number: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §pair item by item. Every lemma cited in this paragraph is stated in the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation for an admissible cylindrical potential $V$, positive $\beta,\kappa$ with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$, and the Gibbs entropy pair with potential $V$ and temperature $\beta$, which are exactly our data.

(i) The pair is noise-closed by The Gibbs Entropy Pair is Noise-Closed, and Its Penalty Bounds the Squared Noise Wasserstein Distance to the Reference Measure §noise-closed.

(ii) It has closed score along noise couplings by The Gibbs Entropy Pair Has Closed Score Along Noise Couplings §closed.

(iii) It has regular penalised maxima by The Gibbs Entropy Pair Has Regular Penalised Maxima §regular.

(iv) Let $\lambda_{c}=\frac{\beta}{\kappa}-K\in\mathbb{R}$. Since $K$ is nonnegative and $V$ has semiconvexity constant $K$, The Gibbs Entropy Pair of a K-Semiconvex Potential is Lambda-Displacement Convex with Lambda the Temperature over the Variance-to-Noise Bound Minus K §convex shows that the pair is $\lambda_{c}$-displacement convex in the sense of Lambda-Displacement Convexity of a Noise Penalty Pair §convex.

(v) The noise map property. The Entropy Domain of a Diagonal Gaussian Reference Measure Has the Noise Map Property applies: its hypotheses that $c$ is a variance sequence and that $\rho=\gamma_{c}$ hold by A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §gaussian, and its number $\kappa$ is our $\kappa$. So its set $\mathcal{D}_{H}$ of the $\mu\in\mathcal{P}(X)$ of finite relative entropy with respect to $\gamma_{c}$ is a subset of $\mathcal{P}^{a}_{\rho}$ with the noise map property, by The Entropy Domain of a Diagonal Gaussian Reference Measure Has the Noise Map Property §inclusion and The Entropy Domain of a Diagonal Gaussian Reference Measure Has the Noise Map Property §map-property. Every $\mu\in\mathcal{D}$ has finite relative entropy with respect to $\gamma^{V}_{\beta}$ by The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §penalty-domain, hence with respect to $\gamma_{c}$ by Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §entropy; so $\mathcal{D}\subseteq\mathcal{D}_{H}$, and $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ by The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §penalty-domain. The condition of The Noise Map Property of a Set of Probability Measures §map-property is imposed separately on each $\mu$ of the set (for all $\nu\in\mathcal{P}^{a}_{\rho}$), and every $\mu\in\mathcal{D}$ is a member of $\mathcal{D}_{H}$, for which it holds; hence $\mathcal{D}$ has the noise map property.

(vi) The growth condition. By Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §below there is $b_{0}\in\mathbb{R}$ with $-b_{0}\le v(u)$ for every $u\in\mathbb{R}^{d}$, $d$ and $v$ being the head dimension and profile of $V$. Let $Z_{V,\beta}$ be the normaliser of The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §normaliser, a positive real number by that clause, and $\log Z_{V,\beta}$ its natural logarithm, as in The Gibbs Entropy Pair is Noise-Closed, and Its Penalty Bounds the Squared Noise Wasserstein Distance to the Reference Measure. Put

$$K'=\frac{2\kappa}{\beta}\bigl(1+|b_{0}|+\beta\,|\log Z_{V,\beta}|\bigr),$$

a real number. Let $\mu\in\mathcal{D}$. By The Gibbs Entropy Pair is Noise-Closed, and Its Penalty Bounds the Squared Noise Wasserstein Distance to the Reference Measure §growth, applied with $b_{0}$ in place of its $b$,

$$W_{a}(\mu,\rho)^{2}\le\frac{2\kappa}{\beta}\bigl(\mathcal{E}(\mu)+b_{0}-\beta\log Z_{V,\beta}\bigr).$$

Write $s=1+|b_{0}|+\beta|\log Z_{V,\beta}|$, so that $s\ge1$. Then $\mathcal{E}(\mu)+b_{0}-\beta\log Z_{V,\beta}\le|\mathcal{E}(\mu)|+|b_{0}|+\beta|\log Z_{V,\beta}|\le s\,|\mathcal{E}(\mu)|+s=s\,(1+|\mathcal{E}(\mu)|)$, using $|\mathcal{E}(\mu)|\le s|\mathcal{E}(\mu)|$ as $s\ge1$. Multiplying by the positive number $\frac{2\kappa}{\beta}$,

$$W_{a}(\mu,\rho)^{2}\le K'\bigl(1+|\mathcal{E}(\mu)|\bigr)\qquad\text{for every }\mu\in\mathcal{D},$$

which is the growth condition of Well-Posedness of the Hamilton-Jacobi Equation with Penalty Drift for a Lambda-Displacement Convex Noise Penalty Pair, Lambda Any Real Number: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §pair with $K'$ in place of its $K$.

So the hypothesis (The pair) holds.

The hypothesis (The data). The conditions imposed on $\lambda_{0}$, $\theta$ and $g$ in Well-Posedness of the Hamilton-Jacobi Equation with Penalty Drift for a Lambda-Displacement Convex Noise Penalty Pair, Lambda Any Real Number: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §data are those of the corollary's clause (The data): $0<\lambda_{0}$, $0<\theta\le1$, and $g:\mathcal{D}\to\mathbb{R}$ bounded and uniformly continuous on $\mathcal{D}$ relative to $\mathcal{D}$ in $(\mathcal{P}^{a}_{\rho},W_{a})$ with $\mathbb{R}$ carrying the metric of The Absolute Value Metric on the Real Line, the metric space being the same as noted above. The number $C$ fixed in the corollary satisfies $0\le C$ and $|g(\mu)|\le C$ for every $\mu\in\mathcal{D}$, so it may be taken as the number $C$ fixed in the theorem.

The viscosity notions agree. By The Hamilton-Jacobi Equation with Gibbs Score Drift on a Hilbert Space §equation, a viscosity subsolution, supersolution or solution in the sense of the corollary is a function $u:\mathcal{D}\to\mathbb{R}$ that is a viscosity subsolution, supersolution or solution of $F$ relative to the Gibbs entropy pair 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 or Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §solution respectively, where $F$ is the Hamilton-Jacobi operator with Gibbs score drift with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$. By The Hamilton-Jacobi Equation with Gibbs Score Drift on a Hilbert Space §operator, this $F$ is the Hamilton-Jacobi operator with penalty drift of The Discounted Hamilton-Jacobi Equation with a Penalty Drift on the Noise Wasserstein Space §operator for the Gibbs entropy pair with the same $\lambda_{0}$, $\theta$ and $g$. On the other hand, the viscosity notions of Well-Posedness of the Hamilton-Jacobi Equation with Penalty Drift for a Lambda-Displacement Convex Noise Penalty Pair, Lambda Any Real Number: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution are those of the Hamilton-Jacobi equation with penalty drift of this pair with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$, which by The Discounted Hamilton-Jacobi Equation with a Penalty Drift on the Noise Wasserstein Space §equation are the viscosity subsolutions, supersolutions and solutions of that same operator $F$ relative to the same pair 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 respectively. So the two sets of notions coincide.

Conclusion. All hypotheses of Well-Posedness of the Hamilton-Jacobi Equation with Penalty Drift for a Lambda-Displacement Convex Noise Penalty Pair, Lambda Any Real Number: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution hold, with the same viscosity notions and the same number $C$. Claim 1 is Well-Posedness of the Hamilton-Jacobi Equation with Penalty Drift for a Lambda-Displacement Convex Noise Penalty Pair, Lambda Any Real Number: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §comparison, applied with the functions $u$ and $w$ of claim 1 in place of its $u$ and $v$ and with the same $b$ and $b'$: its assumptions are exactly those of claim 1, the penalty-subordinate growth of $u$ from above and of $w$ from below being supplied within that clause from the bounds $u\le b$ and $b'\le w$ and the noise-closedness verified in (i); its conclusion $u(\mu)\le w(\mu)$ for every $\mu\in\mathcal{D}$ is that of claim 1. Claim 2 is Well-Posedness of the Hamilton-Jacobi Equation with Penalty Drift for a Lambda-Displacement Convex Noise Penalty Pair, Lambda Any Real Number: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §existence, and claim 3, including its final sentence, is Well-Posedness of the Hamilton-Jacobi Equation with Penalty Drift for a Lambda-Displacement Convex Noise Penalty Pair, Lambda Any Real Number: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §uniqueness.