Proof of Perron's Method for Envelope Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall: Existence and Well-Posedness

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

Step 0 (hypotheses of the auxiliary lemmas and shrinking the range). By Shift Semicontinuity of a Hamiltonian on Phase-Space Noncommutative Laws §semicontinuity, $\mathcal{H}$ is both lower and upper shift-semicontinuous at noise level $\sigma$. Hence all hypotheses of The Supremum of a Bounded Family of Envelope Viscosity Subsolutions is an Envelope Viscosity Subsolution ($\mathcal{D}$ nonempty; absorption of shifts, lower shift semicontinuity and boundedness at zero momentum of $\mathcal{H}$; closed score of $\mathcal{E}$) and of The Bump Construction for Envelope Viscosity Subsolutions (in addition, upper shift semicontinuity of $\mathcal{H}$ and regular plan subgradients of $\mathcal{E}$) are among those of the theorem. Fix $\delta_{\mathrm{sup}}>0$ as the real $\delta_{1}$ of The Supremum of a Bounded Family of Envelope Viscosity Subsolutions is an Envelope Viscosity Subsolution §range and $\delta_{\mathrm{bump}}>0$ as the real $\delta_{1}$ of The Bump Construction for Envelope Viscosity Subsolutions §bump; both depend only on the data of the setting.

Shrinking the range. If $v$ is a subsolution (respectively supersolution) with shift range $\delta''$ and $0<\delta'''\le\delta''$, then $v$ is a subsolution (respectively supersolution) with shift range $\delta'''$: the conditions required in Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §sub (respectively Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §super) for the levels $\delta$ with $0<\delta\le\delta'''$ are among those assumed for the levels $\delta$ with $0<\delta\le\delta''$.

Claim 1 (Perron's method). Let $g_{-}$ be a subsolution with shift range $\delta_{-}$ and $g_{+}$ a supersolution with shift range $\delta_{+}$, both bounded, with $g_{-}\le g_{+}$ on $\Sigma^{2}_{d}$. Put

$$\delta_{*}=\min\{\delta_{-},\delta_{+},\delta_{\mathrm{sup}},\delta_{\mathrm{bump}}\}>0,$$

so that, by shrinking the range, $g_{-}$ is a subsolution and $g_{+}$ a supersolution with shift range $\delta_{*}$. Fix reals $b_{\pm}\ge0$ with $|g_{\pm}|\le b_{\pm}$ and put $b=\max\{b_{-},b_{+}\}$. Let $\mathcal{F}$ be the set of all bounded functions $v:\Sigma^{2}_{d}\to\mathbb{R}$ that are subsolutions of $(\mathrm{E})$ with shift range $\delta_{*}$ and satisfy $g_{-}\le v\le g_{+}$ on $\Sigma^{2}_{d}$. Then $g_{-}\in\mathcal{F}$, so $\mathcal{F}$ is nonempty, and every $v\in\mathcal{F}$ satisfies $-b\le g_{-}\le v\le g_{+}\le b$, that is, $|v|\le b$. For $\zeta\in\Sigma^{2}_{d}$ let $u(\zeta)$ be the least upper bound of the nonempty set $\{v(\zeta):v\in\mathcal{F}\}$, which is bounded above by $g_{+}(\zeta)$ (The Real Numbers: Standing Notation and Background §bounds).

(a) $u\in\mathcal{F}$. Since $0<\delta_{*}\le\delta_{\mathrm{sup}}$, The Supremum of a Bounded Family of Envelope Viscosity Subsolutions is an Envelope Viscosity Subsolution §bound and The Supremum of a Bounded Family of Envelope Viscosity Subsolutions is an Envelope Viscosity Subsolution §subsolution, applied with the range $\delta_{*}$, the bound $b$ and the family $\mathcal{F}$ (whose supremum function is $u$), give $|u|\le b$ on $\Sigma^{2}_{d}$ and that $u$ is a subsolution with shift range $\delta_{*}$. Moreover $g_{-}(\zeta)\le u(\zeta)$ because $g_{-}\in\mathcal{F}$, and $u(\zeta)\le g_{+}(\zeta)$ because $g_{+}(\zeta)$ is an upper bound of the set whose least upper bound is $u(\zeta)$. So $u\in\mathcal{F}$.

(b) $u$ is a supersolution with shift range $\delta_{*}$. Suppose not. Since $0<\delta_{*}\le\delta_{\mathrm{bump}}$, The Bump Construction for Envelope Viscosity Subsolutions §bump, applied with the range $\delta_{*}$, $g=g_{+}$ (bounded, a supersolution with shift range $\delta_{*}$) and $w=u$ (bounded, a subsolution with shift range $\delta_{*}$, and $u\le g_{+}$), gives a bounded $U:\Sigma^{2}_{d}\to\mathbb{R}$ that is a subsolution with shift range $\delta_{*}$, with $u\le U\le g_{+}$ on $\Sigma^{2}_{d}$ and $U(\kappa_{d}(\nu))>u(\kappa_{d}(\nu))$ for some $\nu\in\mathcal{D}$. Then $g_{-}\le u\le U\le g_{+}$, so $U\in\mathcal{F}$, and therefore $U(\kappa_{d}(\nu))\le u(\kappa_{d}(\nu))$ by the definition of $u$ as an upper bound; a contradiction.

By (a), (b) and Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §solution, $u$ is a bounded solution of $(\mathrm{E})$ with shift range $\delta_{*}$, and $g_{-}\le u\le g_{+}$ on $\Sigma^{2}_{d}$. This proves clause 1.

Claim 2 (Existence). By Hamiltonians on Phase-Space Noncommutative Laws that are Bounded at Zero Momentum at Bounded Positions §bounded with $r=R$ fix a real $K$ with $|\mathcal{H}_{M}(X,0)|\le K$ for every tracial W*-probability space $(H,M,\Omega)$ and every $L^{2}$ $d$-tuple $X$ of it with $\mathrm{law}(X)\in\kappa_{d}(\Sigma_{d,R})$, and put $K'=\max\{K,0\}$, so that $K'\ge0$ and the same bound holds with $K'$. By Constant Barriers for Envelope Viscosity Solutions under Regular Plan Subgradients §sub and Constant Barriers for Envelope Viscosity Solutions under Regular Plan Subgradients §super, applied with $K'$ and the shift range $1$ (its hypotheses, $\mathcal{D}$ nonempty and regular plan subgradients, are assumed), the constant function $g_{-}$ with value $-K'/\rho$ is a subsolution and the constant function $g_{+}$ with value $K'/\rho$ a supersolution of $(\mathrm{E})$, each with shift range $1$. Both are bounded (by $K'/\rho$), and $g_{-}\le g_{+}$ because $K'\ge0$ and $\rho>0$. Claim 1 now gives a bounded solution $u$ of $(\mathrm{E})$ with some shift range. This proves clause 2.

Claim 3 (Uniqueness) and Claim 4 (Continuity). Under the additional assumptions, the hypotheses of Comparison, Uniqueness and Continuity on Energy Sublevel Sets for Envelope Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall hold: $\mathcal{D}$ is nonempty, and $\mathcal{H}$ satisfies the structure condition at bounded positions, is uniformly continuous in the momentum at bounded positions and absorbs shifts at noise level $\sigma$. Hence Comparison, Uniqueness and Continuity on Energy Sublevel Sets for Envelope Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §uniqueness gives $u(\kappa_{d}(\mu))=v(\kappa_{d}(\mu))$ for every $\mu\in\mathcal{D}$ and any two bounded solutions $u,v$, each with some shift range, which is clause 3; and Comparison, Uniqueness and Continuity on Energy Sublevel Sets for Envelope Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §continuity gives, for every bounded solution $u$ with some shift range and every real $c$, the continuity of $\mu\mapsto u(\kappa_{d}(\mu))$ on $\{\mu\in\mathcal{D}:\mathcal{E}(\mu)\le c\}$ for the metric $W_{2}$ of The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §metric, which is clause 4.