Proof of Envelope Viscosity Solutions versus Free-Energy-Penalised Viscosity Solutions: Envelopes of Semicontinuous Functions, and Semicontinuous Envelope Solutions are Penalised Solutions under Absorption

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

Throughout, $\mathcal{D}$ is nonempty and $u:\Sigma^{2}_{d}\to\mathbb{R}$ is bounded. As in The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §metric, $\mathcal{D}\subseteq\Sigma_{d,R}$ by The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §bounds, and semicontinuity of real functions on $\mathcal{D}$ refers to the metric space $(\Sigma_{d,R},W_{2})$. Write $f:\mathcal{D}\to\mathbb{R}$ for the function $f(\mu)=u(\kappa_{d}(\mu))$. By The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §envelopes, for real $\delta>0$ the penalty envelopes are the functions $u^{-}_{\delta}$ and $u^{+}_{\delta}$ of The Penalty Envelopes of a Bounded Function on the Domain of the Wall-Confined Free Energy §upper and The Penalty Envelopes of a Bounded Function on the Domain of the Wall-Confined Free Energy §lower, taken with $S=\mathcal{D}$ in $(\Sigma_{d,R},W_{2})$.

Claim 1 (Upper envelope of an upper semicontinuous function). If $f$ is upper semicontinuous on $\mathcal{D}$, then $u^{-}_{\delta}(\mu)=f(\mu)-\delta\,\mathcal{E}(\mu)$ for every real $\delta>0$ and every $\mu\in\mathcal{D}$.

Let $\delta>0$ be real. By Sublevel Sets of the Wall-Confined Free Energy are Closed for the Wasserstein Distance §lsc, $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $(\Sigma_{d,R},W_{2})$; since $\delta\ge0$, claim 3 of Sums and Nonnegative Multiples of Semicontinuous Functions, applied at every point of $\mathcal{D}$, shows that $\delta\,\mathcal{E}$ is lower semicontinuous at every point of $\mathcal{D}$ relative to $\mathcal{D}$. Since $f$ is upper semicontinuous at every point of $\mathcal{D}$ relative to $\mathcal{D}$, the difference clause (claim 3) of Negation, Restriction, and Separated Differences of Semicontinuous Functions shows that $g=f-\delta\,\mathcal{E}$ is upper semicontinuous at every point of $\mathcal{D}$ relative to $\mathcal{D}$, that is, $g$ is upper semicontinuous on $\mathcal{D}$ in the sense of the definition. By The Penalty Envelopes of a Bounded Function on the Domain of the Wall-Confined Free Energy §upper, $g$ is bounded above near each point of $\mathcal{D}$ and $u^{-}_{\delta}=g^{*}$. As $\mathcal{D}$ is nonempty, the fixed-point clause Properties of the Upper Semicontinuous Envelope §fixed applies to $g$ on $S=\mathcal{D}$ and gives $g^{*}(\mu)=g(\mu)$ for every $\mu\in\mathcal{D}$. Hence $u^{-}_{\delta}(\mu)=f(\mu)-\delta\,\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$. This proves the first assertion of clause 1 of the lemma.

Claim 2 (Lower envelope of a lower semicontinuous function). If $f$ is lower semicontinuous on $\mathcal{D}$, then $u^{+}_{\delta}(\mu)=f(\mu)+\delta\,\mathcal{E}(\mu)$ for every real $\delta>0$ and every $\mu\in\mathcal{D}$.

Let $\delta>0$ be real. As in Claim 1, $\delta\,\mathcal{E}$ is lower semicontinuous at every point of $\mathcal{D}$ relative to $\mathcal{D}$, by Sublevel Sets of the Wall-Confined Free Energy are Closed for the Wasserstein Distance §lsc and claim 3 of Sums and Nonnegative Multiples of Semicontinuous Functions. By the first half of claim 3 of Sums and Nonnegative Multiples of Semicontinuous Functions, applied at every point of $\mathcal{D}$, the sum $h=f+\delta\,\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$. By The Penalty Envelopes of a Bounded Function on the Domain of the Wall-Confined Free Energy §lower, $h$ is bounded below near each point of $\mathcal{D}$ and $u^{+}_{\delta}=h_{*}$, and the fixed-point clause Properties of the Lower Semicontinuous Envelope, by Duality §fixed, applied to $h$ on the nonempty set $S=\mathcal{D}$, gives $h_{*}(\mu)=h(\mu)$ for every $\mu\in\mathcal{D}$. Hence $u^{+}_{\delta}(\mu)=f(\mu)+\delta\,\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$. This proves the second assertion of clause 1 of the lemma.

Claim 3 (Realisation of a bounded plan). Let $\mu\in\mathcal{D}\cap\mathcal{D}_{\Xi}$ and let $\pi$ be a bounded plan at $\mu$. Let $(\mathcal{H}_{\pi},\mathcal{M}_{\pi},\Omega_{\pi})$ be the tracial W*-probability space of The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star, let $X_{\pi}$ and $P_{\pi}$ be the $L^{2}$ $d$-tuples of The Shift of a Bounded Plan by a Self-Adjoint Field §tuples, and put $Q=V^{1}_{\pi}\Xi(\mu)$ and $Q'=(-1)Q$. Then: (a) $\mathrm{law}(X_{\pi},P_{\pi})=\kappa_{2d}(\pi)$; (b) $\mathrm{law}(X_{\pi})=\kappa_{d}(\mu)\in\kappa_{d}(\Sigma_{d,R})$; (c) $\lVert P_{\pi}\rVert_{2}=|\pi|_{\mathrm{mom}}$; (d) $Q$ and $Q'$ are $L^{2}$ $d$-tuples of $(\mathcal{H}_{\pi},\mathcal{M}_{\pi},\Omega_{\pi})$ with $\lVert Q'\rVert_{2}=\lVert Q\rVert_{2}=\lVert\Xi(\mu)\rVert_{2}$; (e) for every real $\delta$,

$$\mathcal{H}_{\mathcal{M}_{\pi}}(X_{\pi},P_{\pi})=\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr),\qquad\mathcal{H}_{\mathcal{M}_{\pi}}(X_{\pi},P_{\pi}+\delta Q)=\mathcal{H}\bigl(\pi\oplus\delta\,\Xi(\mu)\bigr),\qquad\mathcal{H}_{\mathcal{M}_{\pi}}(X_{\pi},P_{\pi}+\delta Q')=\mathcal{H}\bigl(\pi\oplus(-\delta)\,\Xi(\mu)\bigr).$$

By Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans, $\pi\in\Sigma_{2d}$ and $\pi\circ\iota^{1}=\mu$; by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law, $\pi\in\Sigma_{2d,r'}$ for some real $r'>0$, so Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation applies to $\pi$.

(a) For $i\in[2d]$ one has $x_{i}^{*}=x_{i}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, so the operator $L_{x_{i}}$ on $\mathcal{H}_{\pi}$ 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, and it belongs to $\mathcal{M}_{\pi}$ by The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §star-algebra. Thus $s=(L_{x_{1}},\dots,L_{x_{2d}})$ is a self-adjoint $2d$-tuple in $\mathcal{M}_{\pi}$. Its law satisfies $\lambda_{s}(p)=\langle\Omega_{\pi},p(s)\Omega_{\pi}\rangle=\pi(p)$ for every $p\in\mathcal{P}_{2d}$ by The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §law, so $\lambda_{s}=\pi$. Its vacuum tuple is $s\Omega_{\pi}=(\widehat{x_{1}},\dots,\widehat{x_{2d}})$, because $L_{x_{i}}\Omega_{\pi}=\widehat{x_{i}}$ 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; by the definition of the pair in Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations, this $2d$-tuple is exactly $(X_{\pi},P_{\pi})$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded,

$$\mathrm{law}(X_{\pi},P_{\pi})=\mathrm{law}(s\Omega_{\pi})=\kappa_{2d}(\lambda_{s})=\kappa_{2d}(\pi).$$

(b) Let $\mathrm{pr}^{1}=(P^{1},0)$ be the marginal datum of Affine Data and Affine Substitutions of Noncommutative Polynomials §coordinate, and write $Z=(X_{\pi},P_{\pi})$, so $Z_{i}=\widehat{x_{i}}$ for $i\in[2d]$. By the definition of affine images in Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations, for $i\in[d]$

$$(\mathrm{pr}^{1}Z)_{i}=0\cdot\Omega_{\pi}+\sum_{j=1}^{2d}P^{1}_{ij}Z_{j}=Z_{i}=\widehat{x_{i}},$$

so $\mathrm{pr}^{1}Z=X_{\pi}$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward and (a), $\mathrm{law}(X_{\pi})=\mathrm{pr}^{1}_{\#}\mathrm{law}(Z)=\mathrm{pr}^{1}_{\#}\kappa_{2d}(\pi)$; 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 this equals $\kappa_{d}(\pi\circ\sigma_{\mathrm{pr}^{1}})$, and $\sigma_{\mathrm{pr}^{1}}=\iota^{1}$ by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §coordinate (these are the identities recorded in The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §pairings). Hence

$$\mathrm{law}(X_{\pi})=\kappa_{d}(\pi\circ\iota^{1})=\kappa_{d}(\mu).$$

Since $\mu\in\mathcal{D}\subseteq\Sigma_{d,R}$ by The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §bounds, $\mathrm{law}(X_{\pi})\in\kappa_{d}(\Sigma_{d,R})$.

(c) By Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples, $\lVert P_{\pi}\rVert_{2}$ is the nonnegative square root of $\sum_{j=1}^{d}\lVert\widehat{x_{d+j}}\rVert^{2}$. 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 $x_{d+j}^{*}=x_{d+j}$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint),

$$\lVert\widehat{x_{d+j}}\rVert^{2}=\langle\widehat{x_{d+j}},\widehat{x_{d+j}}\rangle=\pi(x_{d+j}^{*}x_{d+j})=\pi(x_{d+j}x_{d+j})\qquad(j\in[d]).$$

So $\lVert P_{\pi}\rVert_{2}$ and the momentum norm $|\pi|_{\mathrm{mom}}$ of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans are nonnegative square roots of the same real number $\sum_{j=1}^{d}\pi(x_{d+j}x_{d+j})$, and they are equal by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root.

(d) By The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §shifts, $\Xi(\mu)=(\zeta_{1},\dots,\zeta_{d})$ is an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\mu},\mathcal{M}_{\mu},\Omega_{\mu})$. Since $\pi\circ\iota^{1}=\mu$, the marginal isometry $V=V^{1}_{\pi}$ of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §isometries maps $\mathcal{H}_{\mu}$ to $\mathcal{H}_{\pi}$, and $Q=V\Xi(\mu)=(V\zeta_{1},\dots,V\zeta_{d})$ is 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. By Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry, $V^{*}V=I$, so by the defining property of the adjoint

$$\lVert V\zeta_{j}\rVert^{2}=\langle V\zeta_{j},V\zeta_{j}\rangle=\langle\zeta_{j},V^{*}V\zeta_{j}\rangle=\lVert\zeta_{j}\rVert^{2}\qquad(j\in[d]).$$

Summing over $j$, $\lVert Q\rVert_{2}^{2}=\lVert\Xi(\mu)\rVert_{2}^{2}$ by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples, so $\lVert Q\rVert_{2}=\lVert\Xi(\mu)\rVert_{2}$ by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root. The real multiple $Q'=(-1)Q=(-V\zeta_{1},\dots,-V\zeta_{d})$ is an $L^{2}$ $d$-tuple by Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §operations, and $\lVert-V\zeta_{j}\rVert=\lVert V\zeta_{j}\rVert$ for each $j$, so $\lVert Q'\rVert_{2}=\lVert Q\rVert_{2}$.

(e) The conventions of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation are in force by The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §conventions, so by the lift convention Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts and (a),

$$\mathcal{H}_{\mathcal{M}_{\pi}}(X_{\pi},P_{\pi})=\mathcal{H}\bigl(\mathrm{law}(X_{\pi},P_{\pi})\bigr)=\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr).$$

For real $\delta$, the sum $P_{\pi}+\delta Q=P_{\pi}+\delta\,V^{1}_{\pi}\Xi(\mu)$ is formed with the operations of Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §operations, which are those used both in The Shift of a Bounded Plan by a Self-Adjoint Field §shift and, via Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing, in Absorption of Shifts by a Hamiltonian on Phase-Space Noncommutative Laws §absorption; so by the lift convention and The Shift of a Bounded Plan by a Self-Adjoint Field §shift, with $\zeta=\Xi(\mu)$ and $t=\delta$,

$$\mathcal{H}_{\mathcal{M}_{\pi}}(X_{\pi},P_{\pi}+\delta Q)=\mathcal{H}\bigl(\mathrm{law}(X_{\pi},P_{\pi}+\delta\,V^{1}_{\pi}\Xi(\mu))\bigr)=\mathcal{H}\bigl(\pi\oplus\delta\,\Xi(\mu)\bigr).$$

Finally, componentwise $\delta\,(-V\zeta_{j})=(-\delta)\,V\zeta_{j}$, so $P_{\pi}+\delta Q'=P_{\pi}+(-\delta)Q$, and the previous display with $-\delta$ in place of $\delta$ gives the third identity.

Claim 4 (Absorption at the wall radius). Assume that $\mathcal{H}$ absorbs shifts at noise level $\sigma$, as in claim 2 of the statement. Then there are a real $\delta_{1}>0$ and a nondecreasing function $\omega:[0,\infty)\to[0,\infty)$ such that, for every $\mu\in\mathcal{D}\cap\mathcal{D}_{\Xi}$, every bounded plan $\pi$ at $\mu$ and every real $\delta$ with $0<\delta\le\delta_{1}$,

$$\mathcal{H}\bigl(\pi\oplus\delta\,\Xi(\mu)\bigr)\ge\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)-\delta\,\omega\bigl(|\pi|_{\mathrm{mom}}\bigr)-\frac{\sigma^{2}\delta}{4}\lVert\Xi(\mu)\rVert_{2}^{2},$$ $$\mathcal{H}\bigl(\pi\oplus(-\delta)\,\Xi(\mu)\bigr)\le\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)+\delta\,\omega\bigl(|\pi|_{\mathrm{mom}}\bigr)+\frac{\sigma^{2}\delta}{4}\lVert\Xi(\mu)\rVert_{2}^{2}.$$

The Hamiltonian $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$ of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §hamiltonian, with the lifts of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts, absorbs shifts at the noise level $\sigma$ of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §parameters by the assumption of this claim. Apply Absorption of Shifts by a Hamiltonian on Phase-Space Noncommutative Laws §absorption with $r=R$: it provides a real $\delta_{1}>0$ and a nondecreasing $\omega:[0,\infty)\to[0,\infty)$; fix them. Let $\mu$, $\pi$ and $\delta$ be as stated, and use the notation of Claim 3. The space $(\mathcal{H}_{\pi},\mathcal{M}_{\pi},\Omega_{\pi})$ is a tracial W*-probability space, $X_{\pi}$, $P_{\pi}$, $Q$ and $Q'$ are $L^{2}$ $d$-tuples of it (by The Shift of a Bounded Plan by a Self-Adjoint Field §tuples and Claim 3(d)), and $\mathrm{law}(X_{\pi})\in\kappa_{d}(\Sigma_{d,R})$ by Claim 3(b). Hence the absorption inequality applies with $X=X_{\pi}$, $P=P_{\pi}$ and either $Q$ or $Q'$. With $Q$, using Claim 3(c), (d) and (e),

$$\Bigl|\mathcal{H}\bigl(\pi\oplus\delta\,\Xi(\mu)\bigr)-\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)\Bigr|\le\delta\,\omega\bigl(|\pi|_{\mathrm{mom}}\bigr)+\frac{\sigma^{2}\delta}{4}\lVert\Xi(\mu)\rVert_{2}^{2},$$

and with $Q'$ the same bound holds for $\bigl|\mathcal{H}\bigl(\pi\oplus(-\delta)\,\Xi(\mu)\bigr)-\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)\bigr|$, because $\lVert Q'\rVert_{2}=\lVert\Xi(\mu)\rVert_{2}$. Since $|a-b|\le c$ implies $a\ge b-c$ and $a\le b+c$ for real $a,b,c$, the first bound yields the first asserted inequality and the second bound yields the second.

Claim 5 (Semicontinuous envelope subsolutions are penalised subsolutions). Assume, as in Claim 4, that $\mathcal{H}$ absorbs shifts at noise level $\sigma$. If $f$ is upper semicontinuous on $\mathcal{D}$ and $u$ is an envelope viscosity subsolution of $(\mathrm{E})$ with shift range $\delta_{0}$, then $u$ is a free-energy-penalised viscosity subsolution of $(\mathrm{E})$.

Let $\delta_{1}$ and $\omega$ be as in Claim 4 and put $\delta_{0}'=\min(\delta_{0},\delta_{1})>0$. We show that the pair $(\delta_{0}',\omega)$ has the property required in Free-Energy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §sub; $\omega$ is a nondecreasing function $[0,\infty)\to[0,\infty)$ by Claim 4. The data $\rho,\sigma,\mathcal{E},\mathcal{D}_{\Xi},\Xi,\mathcal{H}$, bounded plans, $|\cdot|_{\mathrm{mom}}$, $\mathcal{J}$ and $J^{+}$ there are those of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation and of the plan superjet with slack $0$, which are the ones of The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §data. Let $\varphi:\Sigma^{2}_{d}\to\mathbb{R}$, let $\delta$ be real with $0<\delta\le\delta_{0}'$, let $\mu\in\mathcal{D}$ satisfy

$$f(\nu)-\varphi\bigl(\kappa_{d}(\nu)\bigr)-\delta\,\mathcal{E}(\nu)<f(\mu)-\varphi\bigl(\kappa_{d}(\mu)\bigr)-\delta\,\mathcal{E}(\mu)\qquad\text{for every }\nu\in\mathcal{D}\text{ with }\nu\ne\mu,$$

and let $\pi$ be a bounded plan at $\mu$ with $\kappa_{2d}(\pi)\in J^{+}\varphi(\kappa_{d}(\mu))$. By Claim 1, $u^{-}_{\delta}(\lambda)=f(\lambda)-\delta\,\mathcal{E}(\lambda)$ for every $\lambda\in\mathcal{D}$, so the display says that $u^{-}_{\delta}(\nu)-\varphi(\kappa_{d}(\nu))<u^{-}_{\delta}(\mu)-\varphi(\kappa_{d}(\mu))$ for every $\nu\in\mathcal{D}$ with $\nu\ne\mu$. Since $0<\delta\le\delta_{0}$, Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §sub applies to $\delta$, $\varphi$, $\mu$ and $\pi$: it gives $\mu\in\mathcal{D}_{\Xi}$ and

$$\rho\bigl(u^{-}_{\delta}(\mu)+\delta\,\mathcal{E}(\mu)\bigr)+\mathcal{H}\bigl(\pi\oplus\delta\,\Xi(\mu)\bigr)+\frac{\sigma^{2}}{2}\Bigl(\mathcal{J}\bigl(\Xi(\mu),\pi\bigr)+\delta\,\lVert\Xi(\mu)\rVert_{2}^{2}\Bigr)\le0,$$

where $u^{-}_{\delta}(\mu)+\delta\,\mathcal{E}(\mu)=f(\mu)=u(\kappa_{d}(\mu))$ by Claim 1. Now $\mu\in\mathcal{D}\cap\mathcal{D}_{\Xi}$ and $0<\delta\le\delta_{1}$, so the first inequality of Claim 4 applies. Writing $m=|\pi|_{\mathrm{mom}}$ and $\xi=\lVert\Xi(\mu)\rVert_{2}^{2}$, we obtain

$$\rho\,u\bigl(\kappa_{d}(\mu)\bigr)+\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)+\frac{\sigma^{2}}{2}\mathcal{J}\bigl(\Xi(\mu),\pi\bigr)\le\rho\,u\bigl(\kappa_{d}(\mu)\bigr)+\mathcal{H}\bigl(\pi\oplus\delta\,\Xi(\mu)\bigr)+\delta\,\omega(m)+\frac{\sigma^{2}\delta}{4}\xi+\frac{\sigma^{2}}{2}\mathcal{J}\bigl(\Xi(\mu),\pi\bigr)$$ $$=\Bigl[\rho\,u\bigl(\kappa_{d}(\mu)\bigr)+\mathcal{H}\bigl(\pi\oplus\delta\,\Xi(\mu)\bigr)+\frac{\sigma^{2}}{2}\Bigl(\mathcal{J}\bigl(\Xi(\mu),\pi\bigr)+\delta\,\xi\Bigr)\Bigr]+\delta\,\omega(m)-\frac{\sigma^{2}\delta}{4}\xi\le\delta\,\omega(m)-\frac{\sigma^{2}\delta}{4}\xi\le\delta\,\omega(m),$$

the last step because $\sigma^{2}\delta\,\xi/4\ge0$. This is the inequality of Free-Energy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §sub, together with $\mu\in\mathcal{D}_{\Xi}$. As $\varphi$, $\delta$, $\mu$ and $\pi$ were arbitrary, $u$ is a free-energy-penalised viscosity subsolution of $(\mathrm{E})$; this is the first assertion of clause 2 of the lemma.

Claim 6 (Semicontinuous envelope supersolutions are penalised supersolutions). Assume, as in Claim 4, that $\mathcal{H}$ absorbs shifts at noise level $\sigma$. If $f$ is lower semicontinuous on $\mathcal{D}$ and $u$ is an envelope viscosity supersolution of $(\mathrm{E})$ with shift range $\delta_{0}$, then $u$ is a free-energy-penalised viscosity supersolution of $(\mathrm{E})$.

Let $\delta_{1}$, $\omega$ be as in Claim 4 and $\delta_{0}'=\min(\delta_{0},\delta_{1})$; we verify Free-Energy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §super with $(\delta_{0}',\omega)$. Let $\varphi:\Sigma^{2}_{d}\to\mathbb{R}$, let $0<\delta\le\delta_{0}'$, let $\mu\in\mathcal{D}$ satisfy

$$f(\nu)-\varphi\bigl(\kappa_{d}(\nu)\bigr)+\delta\,\mathcal{E}(\nu)>f(\mu)-\varphi\bigl(\kappa_{d}(\mu)\bigr)+\delta\,\mathcal{E}(\mu)\qquad\text{for every }\nu\in\mathcal{D}\text{ with }\nu\ne\mu,$$

and let $\pi$ be a bounded plan at $\mu$ with $\kappa_{2d}(\pi)\in J^{-}\varphi(\kappa_{d}(\mu))$. By Claim 2, $u^{+}_{\delta}(\lambda)=f(\lambda)+\delta\,\mathcal{E}(\lambda)$ for every $\lambda\in\mathcal{D}$, so $u^{+}_{\delta}(\nu)-\varphi(\kappa_{d}(\nu))>u^{+}_{\delta}(\mu)-\varphi(\kappa_{d}(\mu))$ for every $\nu\in\mathcal{D}$ with $\nu\ne\mu$. Since $0<\delta\le\delta_{0}$, Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §super gives $\mu\in\mathcal{D}_{\Xi}$ and

$$\rho\bigl(u^{+}_{\delta}(\mu)-\delta\,\mathcal{E}(\mu)\bigr)+\mathcal{H}\bigl(\pi\oplus(-\delta)\,\Xi(\mu)\bigr)+\frac{\sigma^{2}}{2}\Bigl(\mathcal{J}\bigl(\Xi(\mu),\pi\bigr)-\delta\,\lVert\Xi(\mu)\rVert_{2}^{2}\Bigr)\ge0,$$

where $u^{+}_{\delta}(\mu)-\delta\,\mathcal{E}(\mu)=u(\kappa_{d}(\mu))$ by Claim 2. As $\mu\in\mathcal{D}\cap\mathcal{D}_{\Xi}$ and $0<\delta\le\delta_{1}$, the second inequality of Claim 4 applies; with $m=|\pi|_{\mathrm{mom}}$ and $\xi=\lVert\Xi(\mu)\rVert_{2}^{2}$,

$$0\le\rho\,u\bigl(\kappa_{d}(\mu)\bigr)+\mathcal{H}\bigl(\pi\oplus(-\delta)\,\Xi(\mu)\bigr)+\frac{\sigma^{2}}{2}\mathcal{J}\bigl(\Xi(\mu),\pi\bigr)-\frac{\sigma^{2}\delta}{2}\xi\le\rho\,u\bigl(\kappa_{d}(\mu)\bigr)+\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)+\frac{\sigma^{2}}{2}\mathcal{J}\bigl(\Xi(\mu),\pi\bigr)+\delta\,\omega(m)-\frac{\sigma^{2}\delta}{4}\xi.$$

Hence

$$\rho\,u\bigl(\kappa_{d}(\mu)\bigr)+\mathcal{H}\bigl(\kappa_{2d}(\pi)\bigr)+\frac{\sigma^{2}}{2}\mathcal{J}\bigl(\Xi(\mu),\pi\bigr)\ge-\delta\,\omega(m)+\frac{\sigma^{2}\delta}{4}\xi\ge-\delta\,\omega(m),$$

since $\sigma^{2}\delta\,\xi/4\ge0$. This, with $\mu\in\mathcal{D}_{\Xi}$, is the inequality of Free-Energy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §super; as $\varphi$, $\delta$, $\mu$ and $\pi$ were arbitrary, $u$ is a free-energy-penalised viscosity supersolution of $(\mathrm{E})$, the second assertion of clause 2 of the lemma.

Claims 1 and 2 are clause 1 of the lemma, and Claims 5 and 6 are clause 2. $\blacksquare$