Proof of Constant Viscosity Subsolutions and Supersolutions of the Hamilton-Jacobi Equation with Penalty Drift on the Noise Wasserstein Space

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

We work in the setting of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation; elementary order and arithmetic of real numbers is carried by The Real Numbers: Standing Notation and Background §background, which we adopt for that purpose. By Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$, and $\mathcal{D}$ is regarded as a subset of 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, with $\mathbb{R}$ carrying the metric of The Absolute Value Metric on the Real Line. For $\nu\in\mathcal{P}^{a}_{\rho}$, $L^{2}(\nu;X^{a})$ is a real Hilbert space by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields; we write $0_{\nu}$ for its zero vector. By The Discounted Hamilton-Jacobi Equation with a Penalty Drift on the Noise Wasserstein Space §operator,

$$F(\nu,r,q)=\lambda_{0}\,r+\frac{\theta}{2}\,\lVert q\rVert_{\nu}^{2}+\langle\Sigma(\nu),q\rangle_{\nu}-g(\nu)\qquad\bigl((\nu,q)\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma}),\ r\in\mathbb{R}\bigr),$$

and its $\delta$-shifts are those of The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts §shifted.

Claim 1 (growth). Since $u_{\kappa}(\mu)\le\kappa$ and $\kappa\le u_{\kappa}(\mu)$ for every $\mu\in\mathcal{D}$, Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §growth, applied to the noise-closed pair with $b=\kappa$, shows that $u_{\kappa}$ has penalty-subordinate growth from above and from below. So for every positive $\delta$ both envelopes $(u_{\kappa})^{-}_{\delta}$ and $(u_{\kappa})^{+}_{\delta}$ of The Delta-Envelopes of a Function on the Domain of a Noise Penalty Pair §minus and The Delta-Envelopes of a Function on the Domain of a Noise Penalty Pair §plus are defined.

Step A (the envelopes of $u_{\kappa}$). Fix a positive $\delta$. The function $u_{\kappa}$ is continuous on $\mathcal{D}$: for $x,y\in\mathcal{D}$ the distance between $u_{\kappa}(y)$ and $u_{\kappa}(x)$ is $|\kappa-\kappa|=0$, which is less than every positive $\varepsilon$, so the defining condition at $x$ holds with any positive radius. The penalty $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ by Basic Properties of a Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances §lsc, the pair being noise-closed. By Claim 1 and Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §exact,

$$(u_{\kappa})^{-}_{\delta}(\mu)=\kappa-\delta\,\mathcal{E}(\mu),\qquad(u_{\kappa})^{+}_{\delta}(\mu)=\kappa+\delta\,\mathcal{E}(\mu)\qquad(\mu\in\mathcal{D}).$$

Step B (an evaluation of $F$). For $\nu\in\mathcal{D}_{\Sigma}$ we have $(\nu,0_{\nu})\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$, and by Elementary Identities in a Real Inner Product Space §zero in the real Hilbert space $L^{2}(\nu;X^{a})$, $\lVert0_{\nu}\rVert_{\nu}=0$ and $\langle\Sigma(\nu),0_{\nu}\rangle_{\nu}=0$, the score $\Sigma(\nu)$ lying in $T^{a}_{\nu}\subseteq L^{2}(\nu;X^{a})$ by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair. Hence

$$F(\nu,\kappa,0_{\nu})=\lambda_{0}\kappa-g(\nu)\qquad(\nu\in\mathcal{D}_{\Sigma}).$$

Step C (diagonal witnesses). Let $\hat{\mu}\in\mathcal{D}_{\Sigma}$, let $\varphi$ be a noise intrinsic test function on $\mathcal{D}$, let $\delta$ and $\varepsilon$ be positive, let $w$ denote either envelope $(u_{\kappa})^{\mp}_{\delta}$, and put $\nu=\hat{\mu}$, $\pi=(\mathrm{id},\mathrm{id})_{\#}\hat{\mu}$, $s=w(\hat{\mu})$ and $q=\nabla\varphi(\hat{\mu})\in L^{2}(\hat{\mu};X^{a})$. Then $\nu\in\mathcal{D}_{\Sigma}$; by A Toolkit for Penalised Comparison on the Noise Wasserstein Space: Constant Test Functions and Linear Combinations of Test Functions, the Identity as a Unique Noise-Optimal Map, and Discrepancies Along the Push-Forward Under (id, id) and Along Glued Couplings §diagonal, $\pi\in\Pi^{a}(\hat{\mu},\hat{\mu})$ with $I^{a}(\pi)=0<\varepsilon^{2}$, and the discrepancy of $q$ and $\nabla\varphi(\hat{\mu})$ along $\pi$ equals $\lVert q-\nabla\varphi(\hat{\mu})\rVert_{\hat{\mu}}^{2}=\lVert0_{\hat{\mu}}\rVert_{\hat{\mu}}^{2}=0<\varepsilon^{2}$ (Elementary Identities in a Real Inner Product Space §zero); and $|w(\nu)-w(\hat{\mu})|=0<\varepsilon$ and $|s-w(\hat{\mu})|=0<\varepsilon$. So these witnesses satisfy the first four requirements of Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §subsolution and of Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §supersolution; it remains to check the last one in each case.

Claim 2 (subsolution). Assume $\lambda_{0}\kappa\le g(\nu)$ for every $\nu\in\mathcal{D}_{\Sigma}$. By Claim 1, $u_{\kappa}$ has penalty-subordinate growth from above. Let $\delta$ with $0<\delta<1$, a noise intrinsic test function $\varphi$ on $\mathcal{D}$, a point $\hat{\mu}\in\mathcal{D}$ at which $(u_{\kappa})^{-}_{\delta}-\varphi$ has a local maximum relative to $\mathcal{D}$, and a positive $\varepsilon$ be given, in this order; the witnesses chosen below do not depend on $\varepsilon$.

Step 1. By Step A, $(u_{\kappa})^{-}_{\delta}-\varphi$ has values $\kappa-\delta\mathcal{E}(\mu)-\varphi(\mu)$; so, by the local maximum hypothesis, there is a positive radius $\varepsilon_{0}$ with $\kappa-\delta\mathcal{E}(y)-\varphi(y)\le\kappa-\delta\mathcal{E}(\hat{\mu})-\varphi(\hat{\mu})$ for every $y\in\mathcal{D}$ with $W_{a}(\hat{\mu},y)<\varepsilon_{0}$. Subtracting $\kappa$, the function $\chi-\delta\mathcal{E}$ with $\chi=-\varphi$, the function on $\mathcal{P}^{a}_{\rho}$ with value $-\varphi(\mu)$ at $\mu$, has a local maximum at $\hat{\mu}$ relative to $\mathcal{D}$. By A Toolkit for Penalised Comparison on the Noise Wasserstein Space: Constant Test Functions and Linear Combinations of Test Functions, the Identity as a Unique Noise-Optimal Map, and Discrepancies Along the Push-Forward Under (id, id) and Along Glued Couplings §linear, applied with $Q=\mathcal{D}$, with $\varphi$ in the role of both its $\varphi$ and its $\psi$, and with $s=-1$ and $t=0$, the function with value $-\varphi(\mu)+0\cdot\varphi(\mu)=-\varphi(\mu)$ at $\mu$, which is $\chi$, is a noise intrinsic test function on $\mathcal{D}$ with $\nabla\chi(\hat{\mu})=-\nabla\varphi(\hat{\mu})+0\cdot\nabla\varphi(\hat{\mu})=-\nabla\varphi(\hat{\mu})$ in the real vector space $L^{2}(\hat{\mu};X^{a})$.

Step 2. Since the pair has regular penalised maxima, Noise Penalty Pairs with Regular Penalised Maxima §regular with $\chi$ and $\lambda=\delta$ gives $\hat{\mu}\in\mathcal{D}_{\Sigma}$. Then The First-Order Condition at a Penalised Extremum of a Noise Intrinsic Test Function on the Noise Wasserstein Space §maximum, applied with $Q=\mathcal{D}$ and this $\chi$, gives $-\nabla\varphi(\hat{\mu})=\delta\Sigma(\hat{\mu})$, hence $\nabla\varphi(\hat{\mu})=-\delta\Sigma(\hat{\mu})$ in $L^{2}(\hat{\mu};X^{a})$.

Step 3. Take the witnesses of Step C with $w=(u_{\kappa})^{-}_{\delta}$, so $s=\kappa-\delta\mathcal{E}(\hat{\mu})$ by Step A. By The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts §shifted, Step 2 and Step B, and the hypothesis at $\hat{\mu}\in\mathcal{D}_{\Sigma}$,

$$F^{-}_{\delta}(\nu,s,q)=F\bigl(\hat{\mu},\ \kappa,\ -\delta\Sigma(\hat{\mu})+\delta\Sigma(\hat{\mu})\bigr)=F(\hat{\mu},\kappa,0_{\hat{\mu}})=\lambda_{0}\kappa-g(\hat{\mu})\le0<\varepsilon .$$

With Step C, all five requirements of Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §subsolution hold, so $u_{\kappa}$ is a viscosity subsolution of $F$ relative to the pair.

Claim 3 (supersolution). Assume $g(\nu)\le\lambda_{0}\kappa$ for every $\nu\in\mathcal{D}_{\Sigma}$. By Claim 1, $u_{\kappa}$ has penalty-subordinate growth from below. Let $\delta$ with $0<\delta<1$, a noise intrinsic test function $\varphi$ on $\mathcal{D}$, a point $\hat{\mu}\in\mathcal{D}$ at which $(u_{\kappa})^{+}_{\delta}-\varphi$ has a local minimum relative to $\mathcal{D}$, and a positive $\varepsilon$ be given, in this order; again the witnesses do not depend on $\varepsilon$.

Step 1. By the local minimum hypothesis and Step A there is a positive radius $\varepsilon_{0}$ with $\kappa+\delta\mathcal{E}(\hat{\mu})-\varphi(\hat{\mu})\le\kappa+\delta\mathcal{E}(y)-\varphi(y)$ for every $y\in\mathcal{D}$ with $W_{a}(\hat{\mu},y)<\varepsilon_{0}$; rearranging, $\varphi(y)-\delta\mathcal{E}(y)\le\varphi(\hat{\mu})-\delta\mathcal{E}(\hat{\mu})$, so $\varphi-\delta\mathcal{E}$ has a local maximum at $\hat{\mu}$ relative to $\mathcal{D}$.

Step 2. By Noise Penalty Pairs with Regular Penalised Maxima §regular with $\chi=\varphi$ and $\lambda=\delta$, $\hat{\mu}\in\mathcal{D}_{\Sigma}$, and The First-Order Condition at a Penalised Extremum of a Noise Intrinsic Test Function on the Noise Wasserstein Space §maximum with $Q=\mathcal{D}$ and $\chi=\varphi$ gives $\nabla\varphi(\hat{\mu})=\delta\Sigma(\hat{\mu})$.

Step 3. Take the witnesses of Step C with $w=(u_{\kappa})^{+}_{\delta}$, so $s=\kappa+\delta\mathcal{E}(\hat{\mu})$ by Step A. By The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts §shifted, Step 2 and Step B, and the hypothesis at $\hat{\mu}\in\mathcal{D}_{\Sigma}$,

$$-\varepsilon<0\le\lambda_{0}\kappa-g(\hat{\mu})=F(\hat{\mu},\kappa,0_{\hat{\mu}})=F\bigl(\hat{\mu},\ \kappa,\ \delta\Sigma(\hat{\mu})-\delta\Sigma(\hat{\mu})\bigr)=F\bigl(\hat{\mu},\ s-\delta\mathcal{E}(\hat{\mu}),\ q-\delta\Sigma(\hat{\mu})\bigr)=F^{+}_{\delta}(\nu,s,q),$$

where $s-\delta\mathcal{E}(\hat{\mu})=\kappa$, $q=\nabla\varphi(\hat{\mu})=\delta\Sigma(\hat{\mu})$ and $\nu=\hat{\mu}$. With Step C, all five requirements of Viscosity Subsolution, Supersolution and Solution of a First-Order Equation on the Noise Wasserstein Space Relative to a Noise Penalty Pair §supersolution hold, so $u_{\kappa}$ is a viscosity supersolution of $F$ relative to the pair. $\blacksquare$