Proof of Constant Barriers for Envelope Viscosity Solutions under Regular Plan Subgradients

Each result cited is universally quantified over the data in its own statement. By The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §metric, $\mathcal{D}\subseteq\Sigma_{d,R}$, and semicontinuity on $\mathcal{D}$ refers to $(\Sigma_{d,R},W_{2})$. For $L^{2}$ $d$-tuples of one tracial W*-probability space, sums and real multiples are the vector operations of $H^{d}$ and $\langle\cdot,\cdot\rangle_{2}$ is the real inner product of $H^{d}$ with norm $\lVert\cdot\rVert_{2}$ (Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §operations, Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing). Lifts satisfy $f_{M}(Y)=f(\mathrm{law}(Y))$ (Lifts of Functions on Square-Integrable Noncommutative Laws to Square-Integrable Tuples §lift, Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts).

Step 0 (preliminaries). (P1) Envelopes of a constant. Let $c$ be real and $u:\Sigma^{2}_{d}\to\mathbb{R}$ the constant function with value $c$; it is bounded. The function $\mu\mapsto u(\kappa_{d}(\mu))=c$ on $\mathcal{D}$ is upper and lower semicontinuous on $\mathcal{D}$ directly from Upper Semicontinuous Function on a Subset of a Metric Space and Lower Semicontinuous Function on a Subset of a Metric Space (with any $\delta>0$, since $c<c+\varepsilon$ and $c-\varepsilon<c$). Hence, by Envelope Viscosity Solutions versus Free-Energy-Penalised Viscosity Solutions: Envelopes of Semicontinuous Functions, and Semicontinuous Envelope Solutions are Penalised Solutions under Absorption §envelopes-upper and Envelope Viscosity Solutions versus Free-Energy-Penalised Viscosity Solutions: Envelopes of Semicontinuous Functions, and Semicontinuous Envelope Solutions are Penalised Solutions under Absorption §envelopes-lower, for every real $\delta>0$ and every $\mu\in\mathcal{D}$,

$$u^{-}_{\delta}(\mu)=c-\delta\,\mathcal{E}(\mu),\qquad u^{+}_{\delta}(\mu)=c+\delta\,\mathcal{E}(\mu).$$

(P2) The GNS realisation of a bounded plan. Let $\mu\in\mathcal{D}$ and let $\pi$ be a bounded plan at $\mu$ (Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans), so $\pi\in\Sigma_{2d}$ and $\pi\circ\iota^{1}=\mu$. By The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star, $(\mathcal{H}_{\pi},\mathcal{M}_{\pi},\Omega_{\pi})$ is a tracial W*-probability space with $\mathcal{M}_{\pi}=\mathcal{A}_{\pi}''$. Each multiplication operator $L_{x_{i}}$ ($i\in[2d]$, Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §multiplication) lies in $\mathcal{A}_{\pi}\subseteq\mathcal{A}_{\pi}''=\mathcal{M}_{\pi}$ (The Commutant of a Set of Bounded Operators on a Complex Hilbert Space §commutant) and is self-adjoint by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §adjoint, since $x_{i}^{*}=x_{i}$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint). So $L=(L_{x_{1}},\dots,L_{x_{2d}})$ is a self-adjoint $2d$-tuple in $\mathcal{M}_{\pi}$, whose law is $\lambda_{L}=\pi$ by The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §law and Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §law, and whose vacuum tuple is $(\widehat{x_{1}},\dots,\widehat{x_{2d}})=(X_{\pi},P_{\pi})$ by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §vacuum and The Shift of a Bounded Plan by a Self-Adjoint Field §tuples. Hence $\mathrm{law}(X_{\pi},P_{\pi})=\kappa_{2d}(\pi)$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded. Since $\mathrm{pr}^{1}(X_{\pi},P_{\pi})=X_{\pi}$ (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations), Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §bounded and Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §coordinate give $\mathrm{law}(X_{\pi})=\mathrm{pr}^{1}_{\#}\kappa_{2d}(\pi)=\kappa_{d}(\pi\circ\iota^{1})=\kappa_{d}(\mu)\in\kappa_{d}(\Sigma_{d,R})$. Consequently $|\mathcal{H}_{\mathcal{M}_{\pi}}(X_{\pi},0)|\le K$ by the hypothesis on $K$.

(P3) Rescaled plans. For a real $t\ne0$ let $T_{t}=(A^{t},0)$ be the affine datum from $2d$ to $2d$ variables with $A^{t}_{ii}=1$ and $A^{t}_{d+i,d+i}=t$ for $i\in[d]$ and all other entries $0$. By the composite formula of Affine Data and Affine Substitutions of Noncommutative Polynomials §composite, $T_{s}\circ T_{t}=T_{st}$, $T_{1}=\mathrm{id}_{2d}$ and $\mathrm{pr}^{1}\circ T_{t}=\mathrm{pr}^{1}$. For a bounded plan $\pi$ at $\mu$, put $\pi_{t}=\pi\circ\sigma_{T_{t}}$. Then $\pi_{t}\in\Sigma_{2d}$ by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §self-adjoint, and $\pi_{t}\circ\iota^{1}=(\pi\circ\sigma_{T_{t}})\circ\sigma_{\mathrm{pr}^{1}}=\pi\circ\sigma_{\mathrm{pr}^{1}\circ T_{t}}=\pi\circ\iota^{1}=\mu$ by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §coordinate and Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §composition; so $\pi_{t}$ is a bounded plan at $\mu$. By Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §bounded, $\kappa_{2d}(\pi_{t})=(T_{t})_{\#}\kappa_{2d}(\pi)$, and $(T_{1/t})_{\#}\kappa_{2d}(\pi_{t})=\kappa_{2d}(\pi_{t}\circ\sigma_{T_{1/t}})=\kappa_{2d}(\pi\circ\sigma_{T_{1}})=\kappa_{2d}(\pi)$ by the same clause and Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §composition. Moreover, by (P2) and Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, $\mathrm{law}(X_{\pi},tP_{\pi})=\mathrm{law}(T_{t}(X_{\pi},P_{\pi}))=(T_{t})_{\#}\kappa_{2d}(\pi)=\kappa_{2d}(\pi_{t})$, the entries of $T_{t}(X_{\pi},P_{\pi})$ being those of $(X_{\pi},tP_{\pi})$ (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations).

(P4) Rescaled jets. Let $\varphi:\Sigma^{2}_{d}\to\mathbb{R}$, $\mu\in\mathcal{D}$ and $\pi$ a bounded plan at $\mu$; functions $-\varphi/\delta$ and $\varphi/\delta$ are taken pointwise on $\Sigma^{2}_{d}$.

(P4a) Let $\delta>0$, $t=-1/\delta$, $\chi=-\varphi/\delta$, and suppose $\kappa_{2d}(\pi)\in J^{+}\varphi(\kappa_{d}(\mu))$. First, $\kappa_{2d}(\pi_{t})$ is a plan at $\kappa_{d}(\mu)$ (Plans at a Square-Integrable Noncommutative Law §plan), since $\mathrm{pr}^{1}_{\#}\kappa_{2d}(\pi_{t})=\kappa_{d}(\pi_{t}\circ\iota^{1})=\kappa_{d}(\mu)$ by Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §bounded, Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §coordinate and (P3). Let $\eta>0$. By Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §superjet and Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super (slack $0$), applied to $\varphi$ and $\kappa_{2d}(\pi)$ with $\delta\eta$ in place of $\eta$, there is $r>0$ such that $\varphi_{M}(X')\le\varphi(\kappa_{d}(\mu))+\langle P,X'-X\rangle_{2}+\delta\eta\lVert X'-X\rVert_{2}$ whenever $\mathrm{law}(X,P)=\kappa_{2d}(\pi)$ and $\lVert X'-X\rVert_{2}<r$. Let $X,P',X'$ be $L^{2}$ $d$-tuples of a tracial W*-probability space $(H,M,\Omega)$ with $\mathrm{law}(X,P')=\kappa_{2d}(\pi_{t})$ and $\lVert X'-X\rVert_{2}<r$, and put $P=-\delta P'$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward and (P3), $\mathrm{law}(X,P)=\mathrm{law}(T_{-\delta}(X,P'))=(T_{-\delta})_{\#}\kappa_{2d}(\pi_{t})=\kappa_{2d}(\pi)$. So the superdifferential inequality holds for $(X,P,X')$; multiplying it by $-1/\delta<0$ and using $\chi_{M}(X')=-\varphi_{M}(X')/\delta$ and $\langle-\delta P',X'-X\rangle_{2}=-\delta\langle P',X'-X\rangle_{2}$ gives

$$\chi_{M}(X')\ge\chi(\kappa_{d}(\mu))+\langle P',X'-X\rangle_{2}-\eta\lVert X'-X\rVert_{2}.$$

Hence $\kappa_{2d}(\pi_{t})\in J^{-}\chi(\kappa_{d}(\mu))$ by Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §sub and Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §subjet.

(P4b) Let $\delta>0$, $t=1/\delta$, $\chi=\varphi/\delta$, and suppose $\kappa_{2d}(\pi)\in J^{-}\varphi(\kappa_{d}(\mu))$. As in (P4a), $\kappa_{2d}(\pi_{t})$ is a plan at $\kappa_{d}(\mu)$; given $\eta>0$, Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §sub with $\delta\eta$ gives $r>0$ with $\varphi_{M}(X')\ge\varphi(\kappa_{d}(\mu))+\langle P,X'-X\rangle_{2}-\delta\eta\lVert X'-X\rVert_{2}$ whenever $\mathrm{law}(X,P)=\kappa_{2d}(\pi)$ and $\lVert X'-X\rVert_{2}<r$. For $\mathrm{law}(X,P')=\kappa_{2d}(\pi_{t})$ put $P=\delta P'$, so $\mathrm{law}(X,P)=(T_{\delta})_{\#}\kappa_{2d}(\pi_{t})=\kappa_{2d}(\pi)$; dividing by $\delta>0$ gives $\chi_{M}(X')\ge\chi(\kappa_{d}(\mu))+\langle P',X'-X\rangle_{2}-\eta\lVert X'-X\rVert_{2}$, so $\kappa_{2d}(\pi_{t})\in J^{-}\chi(\kappa_{d}(\mu))$.

(P5) Identification with the score (Claim R). Let $\mu\in\mathcal{D}_{\Xi}$, let $\pi$ be a bounded plan at $\mu$ and let $t\ne0$ be real with $\kappa_{2d}(\pi_{t})=\pi^{\Xi}_{\mu}$, where $\pi^{\Xi}_{\mu}=\mathrm{law}(X_{\mu},\Xi(\mu))$ is the score plan of The Score Plan of a Law in the Score Domain of the Wall-Confined Free Energy §score-plan. Put $Q=V^{1}_{\pi}\Xi(\mu)$, an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\pi},\mathcal{M}_{\pi},\Omega_{\pi})$ by The Shift of a Bounded Plan by a Self-Adjoint Field §field. Claim R: $tP_{\pi}=Q$. Indeed, by (P2), $X_{\pi}$ and $P_{\pi}$ are $L^{2}$ $d$-tuples of $(\mathcal{H}_{\pi},\mathcal{M}_{\pi},\Omega_{\pi})$ with $\mathrm{law}(X_{\pi},P_{\pi})=\kappa_{2d}(\pi)$ and $\mathrm{law}(X_{\pi})=\kappa_{d}(\mu)$, where $\mu\in\mathcal{D}\subseteq\Sigma_{d,R}$, and $tP_{\pi}$ is an $L^{2}$ $d$-tuple (Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §operations). By (P3) the hypothesis says $\mathrm{law}(X_{\pi},tP_{\pi})=\kappa_{2d}(\pi_{t})=\mathrm{law}(X_{\mu},\Xi(\mu))$. Here $X_{\mu}$, the tuple of classes of the variables in $\mathcal{H}_{\mu}$ (The Score Plan of a Law in the Score Domain of the Wall-Confined Free Energy), is the tuple $X_{\lambda}$ of classes of the variables in Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum for $\lambda=\mu$, and $\Xi(\mu)$ is an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\mu},\mathcal{M}_{\mu},\Omega_{\mu})$. So Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum §joint, applied with $r=R$, $\lambda=\mu$, $\zeta=\Xi(\mu)$, the tracial W*-probability space $(\mathcal{H}_{\pi},\mathcal{M}_{\pi},\Omega_{\pi})$, the tuple $X_{\pi}$ in the role of $X$, the tuple $tP_{\pi}$ in the role of $Q$, the bounded plan $\pi$ and the tuple $P_{\pi}$ in the role of $P$, gives $\mathrm{law}(X_{\pi},P_{\pi},tP_{\pi})=\mathrm{law}(X_{\pi},P_{\pi},V^{1}_{\pi}\Xi(\mu))$. Pushing both sides forward by the affine datum that keeps the first and the third block (Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations) gives $\mathrm{law}(X_{\pi},Q)=\mathrm{law}(X_{\pi},tP_{\pi})=\mathrm{law}(X_{\mu},\Xi(\mu))$. Now Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum §unique, with the same $r$, $\lambda$, $\zeta$, space and $X$, and with $tP_{\pi}$ and $Q$ in the roles of $Q$ and $Q'$, gives $tP_{\pi}=Q$. In addition, $(V^{1}_{\pi})^{*}V^{1}_{\pi}=I$ by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §isometries and Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry, so $\lVert Q\rVert_{2}=\lVert\Xi(\mu)\rVert_{2}$; and by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plan-pairing, The Shift of a Bounded Plan by a Self-Adjoint Field §tuples and Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing (each $\langle Q_{j},\widehat{x_{d+j}}\rangle$ being real),

$$\mathcal{J}(\Xi(\mu),\pi)=\sum_{j=1}^{d}\operatorname{Re}\langle Q_{j},\widehat{x_{d+j}}\rangle=\langle Q,P_{\pi}\rangle_{2}.$$

Step 1 (Claim 1, lower barrier). Let $c=-K/\rho$ and $u$ the constant function with value $c$. Let $0<\delta\le\delta_{0}$, $\varphi:\Sigma^{2}_{d}\to\mathbb{R}$, $\mu\in\mathcal{D}$ with the strict maximum property of Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §sub, and $\pi$ a bounded plan at $\mu$ with $\kappa_{2d}(\pi)\in J^{+}\varphi(\kappa_{d}(\mu))$. By (P1) the strict maximum property reads $c-\delta\mathcal{E}(\nu)-\varphi(\kappa_{d}(\nu))<c-\delta\mathcal{E}(\mu)-\varphi(\kappa_{d}(\mu))$ for $\nu\in\mathcal{D}$, $\nu\ne\mu$. With $\chi=-\varphi/\delta$, dividing by $-\delta<0$ and rearranging, $\mathcal{E}(\nu)-\chi(\kappa_{d}(\nu))>\mathcal{E}(\mu)-\chi(\kappa_{d}(\mu))$ for these $\nu$; with equality for $\nu=\mu$, the inequality $\mathcal{E}(\nu)-\chi(\kappa_{d}(\nu))\ge\mathcal{E}(\mu)-\chi(\kappa_{d}(\mu))$ holds for every $\nu\in\mathcal{D}$, in particular for those with $W_{2}(\nu,\mu)<1$. With $t=-1/\delta$, $\pi_{t}$ is a bounded plan at $\mu$ by (P3) and $\kappa_{2d}(\pi_{t})\in J^{-}\chi(\kappa_{d}(\mu))$ by (P4a). Since $\mathcal{E}$ has regular plan subgradients, Wall-Confined Free Energies with Regular Plan Subgradients §regular (with $\chi$, $\mu$, $r=1$ and the bounded plan $\pi_{t}$) gives $\mu\in\mathcal{D}_{\Xi}$ and $\kappa_{2d}(\pi_{t})=\pi^{\Xi}_{\mu}$. By (P5), $-P_{\pi}/\delta=Q$, i.e. $P_{\pi}+\delta Q=0$. Hence, by The Shift of a Bounded Plan by a Self-Adjoint Field §shift and the lift, $\mathcal{H}(\pi\oplus\delta\,\Xi(\mu))=\mathcal{H}(\mathrm{law}(X_{\pi},0))=\mathcal{H}_{\mathcal{M}_{\pi}}(X_{\pi},0)\le K$ by (P2); and by (P5), $\mathcal{J}(\Xi(\mu),\pi)+\delta\lVert\Xi(\mu)\rVert_{2}^{2}=\langle Q,P_{\pi}\rangle_{2}+\delta\langle Q,Q\rangle_{2}=\langle Q,P_{\pi}+\delta Q\rangle_{2}=0$. With (P1), $\rho(u^{-}_{\delta}(\mu)+\delta\mathcal{E}(\mu))=\rho c=-K$. So the left side of the inequality of Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §sub is at most $-K+K+0=0$. As $\delta,\varphi,\mu,\pi$ were arbitrary, $u$ is an envelope viscosity subsolution of $(\mathrm{E})$ with shift range $\delta_{0}$.

Step 2 (Claim 2, upper barrier). Let $c=K/\rho$ and $u$ the constant function with value $c$. Let $0<\delta\le\delta_{0}$, $\varphi$, $\mu\in\mathcal{D}$ with the strict minimum property of Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §super, and $\pi$ a bounded plan at $\mu$ with $\kappa_{2d}(\pi)\in J^{-}\varphi(\kappa_{d}(\mu))$. By (P1) the strict minimum property reads $c+\delta\mathcal{E}(\nu)-\varphi(\kappa_{d}(\nu))>c+\delta\mathcal{E}(\mu)-\varphi(\kappa_{d}(\mu))$ for $\nu\ne\mu$; with $\chi=\varphi/\delta$, dividing by $\delta>0$ gives $\mathcal{E}(\nu)-\chi(\kappa_{d}(\nu))\ge\mathcal{E}(\mu)-\chi(\kappa_{d}(\mu))$ for every $\nu\in\mathcal{D}$. With $t=1/\delta$, (P3) and (P4b) give a bounded plan $\pi_{t}$ at $\mu$ with $\kappa_{2d}(\pi_{t})\in J^{-}\chi(\kappa_{d}(\mu))$, and Wall-Confined Free Energies with Regular Plan Subgradients §regular (with $\chi$, $\mu$, $r=1$ and the bounded plan $\pi_{t}$, as in Step 1) gives $\mu\in\mathcal{D}_{\Xi}$ and $\kappa_{2d}(\pi_{t})=\pi^{\Xi}_{\mu}$. By (P5), $P_{\pi}/\delta=Q$, i.e. $P_{\pi}+(-\delta)Q=0$. Hence $\mathcal{H}(\pi\oplus(-\delta)\Xi(\mu))=\mathcal{H}_{\mathcal{M}_{\pi}}(X_{\pi},0)\ge-K$ by The Shift of a Bounded Plan by a Self-Adjoint Field §shift and (P2), and $\mathcal{J}(\Xi(\mu),\pi)-\delta\lVert\Xi(\mu)\rVert_{2}^{2}=\langle Q,P_{\pi}-\delta Q\rangle_{2}=0$. With (P1), $\rho(u^{+}_{\delta}(\mu)-\delta\mathcal{E}(\mu))=\rho c=K$, so the left side of the inequality of Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §super is at least $K-K+0=0$. Hence $u$ is an envelope viscosity supersolution of $(\mathrm{E})$ with shift range $\delta_{0}$.