Proof 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

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

We apply Well-Posedness of the Hamilton-Jacobi Equation with Penalty Drift on the Noise Wasserstein Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution to the Gaussian entropy pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$. 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 $\rho=\gamma_{c}\in\mathcal{P}_{2}(X)$ 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 on the Noise Wasserstein Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution makes no use of that letter, 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 Gaussian Entropy Pair is Noise-Closed, and Its Penalty Bounds the Squared Noise Wasserstein Distance to the Reference Measure. The quadruple $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ 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 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 on the Noise Wasserstein Space: 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 positive $\beta,\kappa$ with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$ and for the Gaussian entropy pair with temperature $\beta$, which are exactly our data. The pair is noise-closed by The Gaussian Entropy Pair is Noise-Closed, and Its Penalty Bounds the Squared Noise Wasserstein Distance to the Reference Measure §noise-closed; it has closed score along noise couplings by The Gaussian Entropy Pair Has Closed Score Along Noise Couplings §closed; it has regular penalised maxima by The Gaussian Entropy Pair Has Regular Penalised Maxima §regular; and it is displacement convex by The Gaussian Entropy Pair is Uniformly Displacement Convex, with Modulus the Temperature over the Variance-to-Noise Bound §displacement-convex. For 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 $\kappa$. Its set $\mathcal{D}_{H}$ of the $\mu\in\mathcal{P}(X)$ of finite relative entropy with respect to $\gamma_{c}$ is $\mathcal{D}$, by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain; so $\mathcal{D}$ has the noise map property of The Noise Map Property of a Set of Probability Measures §map-property by The Entropy Domain of a Diagonal Gaussian Reference Measure Has the Noise Map Property §map-property. Finally let $K=\frac{2\kappa}{\beta}$. By The Gaussian Entropy Pair is Noise-Closed, and Its Penalty Bounds the Squared Noise Wasserstein Distance to the Reference Measure §growth, for every $\mu\in\mathcal{D}$

$$W_{a}(\mu,\rho)^{2}\le K\,\bigl(1+|\mathcal{E}(\mu)|\bigr),$$

which is the growth condition. 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 on the Noise Wasserstein Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §data are those of the corollary: $0<\lambda_{0}$, $0<\theta\le1$, 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 Gaussian 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 Gaussian 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 Gaussian score drift with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$. By The Hamilton-Jacobi Equation with Gaussian 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 Gaussian 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 on the Noise Wasserstein Space: 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 on the Noise Wasserstein Space: 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 on the Noise Wasserstein Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §comparison: its assumptions on $u$, $v$, $b$ and $b'$ are exactly those of claim 1, the penalty-subordinate growth of $u$ from above and of $v$ from below being supplied within that clause from the bounds $u\le b$ and $b'\le v$ and the noise-closedness verified above. Claim 2 is Well-Posedness of the Hamilton-Jacobi Equation with Penalty Drift on the Noise Wasserstein Space: 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 on the Noise Wasserstein Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §uniqueness.