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

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

We work in the setting of The Intrinsic Calculus on the Wasserstein Space: 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 Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\mathcal{D}_{\Sigma}$ is nonempty by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty; $\mathcal{D}$ is regarded as a subset of the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$. By The Discounted Hamilton-Jacobi Equation with Common Noise and a Penalty Drift on the Wasserstein Space §operator,

$$F(\nu,r,q,Y)=\lambda_{0}\,r-\frac{1}{2}\,\mathrm{tr}\bigl(\Gamma^{\top}\Gamma Y\bigr)+\frac{\theta}{2}\,\lVert q\rVert_{\nu}^{2}+\langle\Sigma(\nu),q\rangle_{\nu}-g(\nu),$$

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

Claim 1 (growth). By Basic Properties of a Wasserstein-Closed Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets, Bounded Distances, and Coercive Pairs §bounded-below, applied to the Wasserstein-closed pair, there is $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$. Let $\delta$ be positive; then $\delta e_{0}\le\delta\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$. With $C=\kappa-\delta e_{0}$ we get $u_{\kappa}(\mu)=\kappa\le C+\delta\mathcal{E}(\mu)$, which is Penalty-Subordinate Growth of a Function on the Penalty Domain §above; with $C'=-\kappa-\delta e_{0}$ we get $-C'-\delta\mathcal{E}(\mu)=\kappa+\delta e_{0}-\delta\mathcal{E}(\mu)\le\kappa=u_{\kappa}(\mu)$, which is Penalty-Subordinate Growth of a Function on the Penalty Domain §below. So for every positive $\delta$ both envelopes $(u_{\kappa})^{-}_{\delta}$ and $(u_{\kappa})^{+}_{\delta}$ of The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair §minus and The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair §plus are defined.

Step A (the envelopes of $u_{\kappa}$). Fix a positive $\delta$. The functions $u_{\kappa}-\delta\mathcal{E}$ and $u_{\kappa}+\delta\mathcal{E}$ on $\mathcal{D}$ have values $\kappa-\delta\mathcal{E}(\mu)$ and $\kappa+\delta\mathcal{E}(\mu)$. We show that the first is upper semicontinuous and the second lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$. Let $x\in\mathcal{D}$ and let $\varepsilon$ be positive. By Basic Properties of a Wasserstein-Closed Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets, Bounded Distances, and Coercive Pairs §lsc, $\mathcal{E}$ is lower semicontinuous at $x$ relative to $\mathcal{D}$, so for the positive number $\tfrac{\varepsilon}{\delta}$ there is a positive $\eta$ such that every $y\in\mathcal{D}$ with $W_{2}(x,y)<\eta$ satisfies $\mathcal{E}(x)-\tfrac{\varepsilon}{\delta}<\mathcal{E}(y)$. Multiplying by $\delta>0$ gives $\delta\mathcal{E}(x)-\varepsilon<\delta\mathcal{E}(y)$, hence

$$\kappa-\delta\mathcal{E}(y)<\bigl(\kappa-\delta\mathcal{E}(x)\bigr)+\varepsilon\qquad\text{and}\qquad\bigl(\kappa+\delta\mathcal{E}(x)\bigr)-\varepsilon<\kappa+\delta\mathcal{E}(y)$$

for these $y$, which are the defining inequalities of Upper Semicontinuous Function on a Subset of a Metric Space and Lower Semicontinuous Function on a Subset of a Metric Space at $x$ with this $\eta$. By Claim 1, $u_{\kappa}$ has penalty-subordinate growth from above and from below, so by The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair §minus the function $u_{\kappa}-\delta\mathcal{E}$ is bounded above near each point of $\mathcal{D}$, and by The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair §plus the function $u_{\kappa}+\delta\mathcal{E}$ is bounded below near each point of $\mathcal{D}$; these are the hypotheses of Properties of the Upper Semicontinuous Envelope and Properties of the Lower Semicontinuous Envelope, by Duality respectively. By the fixed-point clauses Properties of the Upper Semicontinuous Envelope §fixed and Properties of the Lower Semicontinuous Envelope, by Duality §fixed (for the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ and the nonempty set $\mathcal{D}$), applied to the upper semicontinuous function $u=u_{\kappa}-\delta\mathcal{E}$ and the lower semicontinuous function $u=u_{\kappa}+\delta\mathcal{E}$ just obtained, we get $u^{*}=u$ for $u=u_{\kappa}-\delta\mathcal{E}$ and $u_{*}=u$ for $u=u_{\kappa}+\delta\mathcal{E}$ on $\mathcal{D}$. Hence the envelopes of The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair §minus and The Delta-Envelopes of a Function on the Penalty Domain Relative to a Penalty Pair §plus are

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

Step B (an evaluation of $F$). For $\nu\in\mathcal{D}_{\Sigma}$ let $0_{\nu}$ be the zero vector of $L^{2}(\nu;\mathbb{R}^{d})$, so $(\nu,0_{\nu})\in\mathcal{V}(\mathcal{D}_{\Sigma})$. Each entry of the matrix product $\Gamma^{\top}\Gamma\,0_{d}$ is a finite sum of products having the factor $0$, hence $0$, so $\Gamma^{\top}\Gamma\,0_{d}=0_{d}$, whose trace, a finite sum of zero entries, is $0$. By Elementary Identities in a Real Inner Product Space §zero in the real Hilbert space $L^{2}(\nu;\mathbb{R}^{d})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu, $\lVert0_{\nu}\rVert_{\nu}=0$ and $\langle\Sigma(\nu),0_{\nu}\rangle_{\nu}=0$. Hence

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

Also, by The Hamilton-Jacobi Operator with Common Noise and Penalty Drift is Degenerate Elliptic, $F$ is degenerate elliptic: $F(\nu,r,q,Y)\le F(\nu,r,q,X)$ whenever $X\preceq Y$.

Step C (diagonal witnesses). Let $\hat{\mu}\in\mathcal{D}_{\Sigma}$, let $\varphi$ be an intrinsic test function on $\mathcal{D}$, let $\delta$ be positive and $\varepsilon$ positive, let $w$ denote either envelope $(u_{\kappa})^{\mp}_{\delta}$, and put $\nu=\hat{\mu}$, $\pi=(\mathrm{id},\mathrm{id})_{\#}\hat{\mu}$, $s=w(\hat{\mu})$, $q=\nabla\varphi(\hat{\mu})\in L^{2}(\hat{\mu};\mathbb{R}^{d})$ and $Y=H_{\varphi}(\hat{\mu})\in\mathcal{S}(d)$. Then $\nu\in\mathcal{D}_{\Sigma}$; $\pi\in\Pi(\hat{\mu},\hat{\mu})$ with $I(\pi)=0<\varepsilon^{2}$ by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §pushforward; $|w(\nu)-w(\hat{\mu})|=0<\varepsilon$ and $|s-w(\hat{\mu})|=0<\varepsilon$; by The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §diagonal the discrepancy of $q$ and $\nabla\varphi(\hat{\mu})$ along $\pi$ equals $\lVert q-\nabla\varphi(\hat{\mu})\rVert_{\hat{\mu}}^{2}=\lVert0_{\hat{\mu}}\rVert_{\hat{\mu}}^{2}=0<\varepsilon^{2}$ (using Elementary Identities in a Real Inner Product Space §zero); and $\lVert Y-H_{\varphi}(\hat{\mu})\rVert=d_{\mathcal{S}(d)}(H_{\varphi}(\hat{\mu}),H_{\varphi}(\hat{\mu}))=0<\varepsilon$, where the identity $\lVert Y-H_{\varphi}(\hat{\mu})\rVert=d_{\mathcal{S}(d)}(H_{\varphi}(\hat{\mu}),H_{\varphi}(\hat{\mu}))$ holds by Distance Between Symmetric Real Matrices, and the value is $0$ because $d_{\mathcal{S}(d)}$ is a metric on $\mathcal{S}(d)$ by The Set of Symmetric Real Matrices is a Metric Space and so vanishes on the diagonal by condition 2 of Metric Space. So these witnesses satisfy the first five requirements of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution and of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §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 $0<\delta<1$, an 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.

Step 1. By Step A, $(u_{\kappa})^{-}_{\delta}-\varphi$ has values $\kappa-\delta\mathcal{E}(\mu)-\varphi(\mu)$; so, by the local maximum hypothesis and Step A, there is a positive radius $\rho_{0}$ 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 $W_{2}(\hat{\mu},y)<\rho_{0}$. Subtracting $\kappa$, the function $\chi-\delta\mathcal{E}$ with $\chi=-\varphi$ has a local maximum at $\hat{\mu}$ relative to $\mathcal{D}$. By Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §difference, $\chi=-\varphi$ is an intrinsic test function on $\mathcal{D}$ with $\nabla\chi(\hat{\mu})=-\nabla\varphi(\hat{\mu})$ and $H_{\chi}(\hat{\mu})=-H_{\varphi}(\hat{\mu})$.

Step 2. Since the pair has regular penalised maxima, Penalty Pairs with Regular Penalised Maxima §regular with $\chi$ and $\lambda=\delta$ gives $\hat{\mu}\in\mathcal{D}_{\Sigma}$. Then First- and Second-Order Conditions at a Penalised Extremum of an Intrinsic Test Function on the Wasserstein Space §maximum, applied with $Q=\mathcal{D}$ and this $\chi$, gives $-\nabla\varphi(\hat{\mu})=\delta\Sigma(\hat{\mu})$ and $-H_{\varphi}(\hat{\mu})\preceq\delta H_{\mathcal{E}}(\hat{\mu})$. Hence $\nabla\varphi(\hat{\mu})=-\delta\Sigma(\hat{\mu})$ in $L^{2}(\hat{\mu};\mathbb{R}^{d})$, and adding $H_{\varphi}(\hat{\mu})$ to both sides of the matrix inequality (claim 3 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure; the sums are computed entrywise by Sum of Real Matrices) gives $0_{d}\preceq H_{\varphi}(\hat{\mu})+\delta H_{\mathcal{E}}(\hat{\mu})$.

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

$$F^{-}_{\delta}(\nu,s,q,Y)=F\bigl(\hat{\mu},\ \kappa,\ -\delta\Sigma(\hat{\mu})+\delta\Sigma(\hat{\mu}),\ H_{\varphi}(\hat{\mu})+\delta H_{\mathcal{E}}(\hat{\mu})\bigr)=F\bigl(\hat{\mu},\kappa,0_{\hat{\mu}},H_{\varphi}(\hat{\mu})+\delta H_{\mathcal{E}}(\hat{\mu})\bigr).$$

By degenerate ellipticity (Step B) with $X=0_{d}\preceq H_{\varphi}(\hat{\mu})+\delta H_{\mathcal{E}}(\hat{\mu})$, and then Step B and the hypothesis at $\hat{\mu}\in\mathcal{D}_{\Sigma}$,

$$F^{-}_{\delta}(\nu,s,q,Y)\le F(\hat{\mu},\kappa,0_{\hat{\mu}},0_{d})=\lambda_{0}\kappa-g(\hat{\mu})\le0<\varepsilon .$$

With Step C, all six requirements of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §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 $0<\delta<1$, an 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.

Step 1. By the local minimum hypothesis and Step A there is a positive radius $\rho_{0}$ 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 $W_{2}(\hat{\mu},y)<\rho_{0}$; 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 Penalty Pairs with Regular Penalised Maxima §regular with $\chi=\varphi$ and $\lambda=\delta$, $\hat{\mu}\in\mathcal{D}_{\Sigma}$, and First- and Second-Order Conditions at a Penalised Extremum of an Intrinsic Test Function on the Wasserstein Space §maximum with $Q=\mathcal{D}$ and $\chi=\varphi$ gives $\nabla\varphi(\hat{\mu})=\delta\Sigma(\hat{\mu})$ and $H_{\varphi}(\hat{\mu})\preceq\delta H_{\mathcal{E}}(\hat{\mu})$. Adding $-\delta H_{\mathcal{E}}(\hat{\mu})$ to both sides (claim 3 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure; sums and differences entrywise by Sum of Real Matrices and Difference of Real Matrices) gives $H_{\varphi}(\hat{\mu})-\delta H_{\mathcal{E}}(\hat{\mu})\preceq0_{d}$.

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

$$F^{+}_{\delta}(\nu,s,q,Y)=F\bigl(\hat{\mu},\ \kappa,\ \delta\Sigma(\hat{\mu})-\delta\Sigma(\hat{\mu}),\ H_{\varphi}(\hat{\mu})-\delta H_{\mathcal{E}}(\hat{\mu})\bigr)=F\bigl(\hat{\mu},\kappa,0_{\hat{\mu}},H_{\varphi}(\hat{\mu})-\delta H_{\mathcal{E}}(\hat{\mu})\bigr).$$

By degenerate ellipticity (Step B) with $X=H_{\varphi}(\hat{\mu})-\delta H_{\mathcal{E}}(\hat{\mu})\preceq0_{d}$, and then Step B and the hypothesis at $\hat{\mu}\in\mathcal{D}_{\Sigma}$,

$$-\varepsilon<0\le\lambda_{0}\kappa-g(\hat{\mu})=F(\hat{\mu},\kappa,0_{\hat{\mu}},0_{d})\le F^{+}_{\delta}(\nu,s,q,Y).$$

With Step C, all six requirements of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §supersolution hold, so $u_{\kappa}$ is a viscosity supersolution of $F$ relative to the pair. $\blacksquare$