Proof of Perron's Method on Unitary Laws: the Upper Envelope of a Bounded Family of Subsolutions Is a Subsolution

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

Conventions. The conventions of Unitary Laws with Free Unitary Noise: Standing Data §spaces and Unitary Laws with Free Unitary Noise: Standing Data §test-functions are in force. Viscosity subsolutions are those of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation with the admissible Hamiltonian $H$ of the statement. By The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §inner-product and The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §hilbert, $E_{d}$ is a real Hilbert space with inner product $\langle\cdot,\cdot\rangle_{d}$ and norm $\lVert\cdot\rVert_{d}$, and its distance is $(y,y')\mapsto\lVert y-y'\rVert_{d}$ (Real Hilbert Spaces: Standing Notation and Background §space); The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity, Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$, Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space and Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space are applied with $E_{d}$ as the space and $U=E_{d}$, which is open in itself (Real Hilbert Spaces: Standing Notation and Background §topology); every member of $C^{2}(E_{d})$ belongs to $C^{1}(E_{d})$ by The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c2, and every member of $C^{1}(E_{d})$ is differentiable on $E_{d}$, that is, at every point of $E_{d}$, by The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c1 and Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §on-set. For $\mu,\nu\in\mathcal{L}_{d}$, $d_{\mathcal{L}}(\mu,\nu)=\lVert\iota_{d}(\mu)-\iota_{d}(\nu)\rVert_{d}$ (The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance §distance); $(\mathcal{L}_{d},d_{\mathcal{L}})$ is a metric space (The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §metric), nonempty (Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words §nonempty) and compact (The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §compact). Convergence of real sequences is convergence in $(\mathbb{R},d_{\mathbb{R}})$ (The Real Numbers: Standing Notation and Background §sequences). For $G\in C^{1}(E_{d})$ and $\lambda\in\mathcal{L}_{d}$ put

$$T_{G}(\lambda)=H\bigl(\lambda,DG(\iota_{d}(\lambda))\bigr)-\frac{\beta^{2}}{2}\bigl\langle DG(\iota_{d}(\lambda)),\widehat{\Theta}\lambda\bigr\rangle_{d}-f(\lambda).$$

Preliminary facts. Let $G\in C^{1}(E_{d})$.

(C1) $G\circ\iota_{d}$ is continuous and bounded on $\mathcal{L}_{d}$. $G$ is continuous on $E_{d}$ by Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space §continuous. Let $\lambda\in\mathcal{L}_{d}$ and $\varepsilon>0$; choose $\eta>0$ with $|G(y)-G(\iota_{d}(\lambda))|<\varepsilon$ whenever $\lVert y-\iota_{d}(\lambda)\rVert_{d}<\eta$ (Continuous Map Between Metric Spaces). If $d_{\mathcal{L}}(\lambda,\lambda')<\eta$ then $\lVert\iota_{d}(\lambda')-\iota_{d}(\lambda)\rVert_{d}=d_{\mathcal{L}}(\lambda',\lambda)<\eta$, so $|G(\iota_{d}(\lambda'))-G(\iota_{d}(\lambda))|<\varepsilon$. Thus $G\circ\iota_{d}$ is continuous on $(\mathcal{L}_{d},d_{\mathcal{L}})$, and Extreme Value Theorem on a Compact Subset of a Metric Space (with $X=K=\mathcal{L}_{d}$) gives $\lambda_{\min},\lambda_{\max}\in\mathcal{L}_{d}$ with $G(\iota_{d}(\lambda_{\min}))\le G(\iota_{d}(\lambda))\le G(\iota_{d}(\lambda_{\max}))$ for all $\lambda$; so $C_{G}=\max\{|G(\iota_{d}(\lambda_{\min}))|,|G(\iota_{d}(\lambda_{\max}))|\}$ satisfies $|G(\iota_{d}(\lambda))|\le C_{G}$ for every $\lambda\in\mathcal{L}_{d}$. Moreover $G\circ\iota_{d}$ is lower semicontinuous on $\mathcal{L}_{d}$: $|G(\iota_{d}(\lambda'))-G(\iota_{d}(\lambda))|<\varepsilon$ implies $G(\iota_{d}(\lambda))-\varepsilon<G(\iota_{d}(\lambda'))$ (Lower Semicontinuous Function on a Subset of a Metric Space).

(C2) Sequential continuity. Let $(\lambda_{n})_{n\in\mathbb{N}}$ converge to $\lambda$ in $(\mathcal{L}_{d},d_{\mathcal{L}})$. Then $(G(\iota_{d}(\lambda_{n})))_{n}$ converges to $G(\iota_{d}(\lambda))$ and $(T_{G}(\lambda_{n}))_{n}$ converges to $T_{G}(\lambda)$. Indeed, $\lVert\iota_{d}(\lambda_{n})-\iota_{d}(\lambda)\rVert_{d}=d_{\mathcal{L}}(\lambda_{n},\lambda)$, so $(\iota_{d}(\lambda_{n}))_{n}$ converges to $\iota_{d}(\lambda)$ in $E_{d}$ (Convergent Sequence in a Metric Space). The first assertion follows from (C1) and Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential. The gradient map $DG$ is continuous on $E_{d}$ (The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c1), so by the same lemma $p_{n}=DG(\iota_{d}(\lambda_{n}))$ converges to $p=DG(\iota_{d}(\lambda))$ in $E_{d}$. By claim 1 of Coordinatewise Convergence, Sequential Compactness and Density in a Product Metric Space, $((\lambda_{n},p_{n}))_{n}$ converges to $(\lambda,p)$ in $\mathcal{L}_{d}\times E_{d}$ with the product metric of Admissible Hamiltonians on Unitary Laws §continuous, on which $H$ is continuous by that property, $H$ being admissible; so $(H(\lambda_{n},p_{n}))_{n}$ converges to $H(\lambda,p)$ by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential. By The Free Unitary Heat Generator: Bound, Continuity and Strict Dissipation in the Length-Weighted Gauge §continuity and the same lemma, $(\widehat{\Theta}\lambda_{n})_{n}$ converges to $\widehat{\Theta}\lambda$ in $E_{d}$, so $(\langle p_{n},\widehat{\Theta}\lambda_{n}\rangle_{d})_{n}$ converges to $\langle p,\widehat{\Theta}\lambda\rangle_{d}$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity. As $f$ is continuous (Unitary Laws with Free Unitary Noise: Standing Data §source), $(f(\lambda_{n}))_{n}$ converges to $f(\lambda)$ by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential. Claims 1 and 3 of Arithmetic of Limits of Real Sequences give the second assertion.

Part 1 (Boundedness). Fix $u_{0}\in\mathcal{S}$; it is bounded (Viscosity Sub- and Supersolutions of the Hamilton-Jacobi-Bellman Equation on Unitary Laws §subsolution), with a bound $M_{0}\ge0$ (Bounded Real-Valued Function on a Set). For $\lambda\in\mathcal{L}_{d}$, $w(\lambda)$ is an upper bound of $\{u(\lambda):u\in\mathcal{S}\}$ and the least one, and $M$ is an upper bound, so $-M_{0}\le u_{0}(\lambda)\le w(\lambda)\le M$. Hence $|w(\lambda)|\le C_{w}$ with $C_{w}=\max\{M_{0},|M|\}\ge0$, and $w$ is bounded.

Part 2 (Upper envelope). Semicontinuity and bound. For every $\lambda\in\mathcal{L}_{d}$ we have $w(y)\le C_{w}$ for all $y\in\mathcal{L}_{d}$, so $C_{w}\in A_{w}(\lambda)$ and $w$ is bounded above near each point of $\mathcal{L}_{d}$ (Upper and Lower Semicontinuous Envelopes of a Real-Valued Function §near-bounds); $w^{*}$ is defined (Upper and Lower Semicontinuous Envelopes of a Real-Valued Function §upper). We apply Properties of the Upper Semicontinuous Envelope with the metric space $(\mathcal{L}_{d},d_{\mathcal{L}})$ and $S=\mathcal{L}_{d}$. By Properties of the Upper Semicontinuous Envelope §bounds and Properties of the Upper Semicontinuous Envelope §usc, $w\le w^{*}$ and $w^{*}$ is upper semicontinuous on $\mathcal{L}_{d}$. The constant function $C_{w}$ is upper semicontinuous (as $C_{w}<C_{w}+\varepsilon$ for $\varepsilon>0$) and majorises $w$, so $w^{*}\le C_{w}$ by Properties of the Upper Semicontinuous Envelope §least. Thus $-C_{w}\le w\le w^{*}\le C_{w}$, and $w^{*}$ is bounded.

The data and the order of choices. Let $F\in C^{1}(E_{d})$ and $\lambda_{0}\in\mathcal{L}_{d}$ be such that $w^{*}-F\circ\iota_{d}$ has a local maximum at $\lambda_{0}$ relative to $\mathcal{L}_{d}$: by Local Maximum of a Function Relative to a Subset of a Metric Space there is a real $r>0$ with

$$w^{*}(\lambda)-F(\iota_{d}(\lambda))\le w^{*}(\lambda_{0})-F(\iota_{d}(\lambda_{0}))\qquad(\lambda\in\mathcal{L}_{d},\ d_{\mathcal{L}}(\lambda,\lambda_{0})<r).\tag{LM}$$

By Viscosity Sub- and Supersolutions of the Hamilton-Jacobi-Bellman Equation on Unitary Laws §subsolution we must show $\rho\,w^{*}(\lambda_{0})+T_{F}(\lambda_{0})\le0$. The objects below are chosen in this order: $C_{F}$, $A$ and $F_{1}$ (Step 1); the sequence $(\lambda_{n})$, then for each $n$ the subsolution $u_{n}$, then the point $\mu_{n}$ (Step 2).

Step 1 (A strict global maximum). Let $C_{F}$ be as in (C1) for $G=F$, and put

$$A=1+\frac{2C_{w}+2C_{F}+1}{r^{2}},\qquad\text{so that }A\ge1\text{ and }A\,r^{2}\ge2C_{w}+2C_{F}+1 .$$

Let $q:E_{d}\to\mathbb{R}$, $q(y)=A\lVert y-\iota_{d}(\lambda_{0})\rVert_{d}^{2}=\tfrac{2A}{2}\lVert y-\iota_{d}(\lambda_{0})\rVert_{d}^{2}$. By Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ §quadratic (with $2A$ and $\iota_{d}(\lambda_{0})$ in the roles of $\alpha$ and $y_{0}$), $q\in C^{1}(E_{d})$ and $Dq(y)=2A(y-\iota_{d}(\lambda_{0}))$. Put $F_{1}=F+q$. By Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §sum, $F_{1}\in C^{1}(E_{d})$ and $DF_{1}(y)=DF(y)+2A(y-\iota_{d}(\lambda_{0}))$; in particular $DF_{1}(\iota_{d}(\lambda_{0}))=DF(\iota_{d}(\lambda_{0}))$, since $\iota_{d}(\lambda_{0})-\iota_{d}(\lambda_{0})$ is the zero vector and so is its multiple by $2A$. Hence

$$T_{F_{1}}(\lambda_{0})=T_{F}(\lambda_{0}).\tag{G0}$$

For $\lambda\in\mathcal{L}_{d}$, $F_{1}(\iota_{d}(\lambda))=F(\iota_{d}(\lambda))+A\,d_{\mathcal{L}}(\lambda,\lambda_{0})^{2}$. Put $\Gamma(\lambda)=w^{*}(\lambda)-F_{1}(\iota_{d}(\lambda))$ and $\Gamma_{0}=\Gamma(\lambda_{0})=w^{*}(\lambda_{0})-F(\iota_{d}(\lambda_{0}))\ge-C_{w}-C_{F}$. We claim

$$\Gamma(\lambda)\le\Gamma_{0}\quad(\lambda\in\mathcal{L}_{d}),\qquad\Gamma(\lambda)\le\Gamma_{0}-s^{2}\quad(0<s\le1,\ \lambda\in\mathcal{L}_{d},\ d_{\mathcal{L}}(\lambda,\lambda_{0})\ge s).\tag{ST}$$

Let $\lambda\in\mathcal{L}_{d}$ and $D=d_{\mathcal{L}}(\lambda,\lambda_{0})$. If $D<r$, then by (LM) and $A\ge1$, $\Gamma(\lambda)=w^{*}(\lambda)-F(\iota_{d}(\lambda))-AD^{2}\le\Gamma_{0}-AD^{2}\le\Gamma_{0}-D^{2}$, which is at most $\Gamma_{0}$, and at most $\Gamma_{0}-s^{2}$ when $D\ge s>0$ (Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field). If $D\ge r$, then $AD^{2}\ge Ar^{2}$ and, by the bounds $w^{*}\le C_{w}$ and $|F\circ\iota_{d}|\le C_{F}$,

$$\Gamma(\lambda)\le C_{w}+C_{F}-Ar^{2}\le-C_{w}-C_{F}-1\le\Gamma_{0}-1,$$

which is at most $\Gamma_{0}$, and at most $\Gamma_{0}-s^{2}$ for $0<s\le1$ since then $s^{2}\le1$.

Step 2 (Approximate maximisers). By Properties of the Upper Semicontinuous Envelope §approximation, choose a sequence $(\lambda_{n})_{n\in\mathbb{N}}$ in $\mathcal{L}_{d}$ converging to $\lambda_{0}$ with $(w(\lambda_{n}))_{n}$ converging to $w^{*}(\lambda_{0})$. For each $n\in\mathbb{N}$, the set $\{u(\lambda_{n}):u\in\mathcal{S}\}$ is nonempty, bounded above by $M$, with least upper bound $w(\lambda_{n})$; by claim 3 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ with $\varepsilon=1/n$ choose $u_{n}\in\mathcal{S}$ with

$$u_{n}(\lambda_{n})>w(\lambda_{n})-\tfrac1n .$$

The function $u_{n}$ is upper semicontinuous (Viscosity Sub- and Supersolutions of the Hamilton-Jacobi-Bellman Equation on Unitary Laws §subsolution) and $F_{1}\circ\iota_{d}$ is lower semicontinuous by (C1), so $u_{n}-F_{1}\circ\iota_{d}$ is upper semicontinuous on $\mathcal{L}_{d}$ by claim 3 of Negation, Restriction, and Separated Differences of Semicontinuous Functions. By claim 1 of Semicontinuous Functions Attain Their Extrema on a Compact Set (with $X=K=\mathcal{L}_{d}$) choose $\mu_{n}\in\mathcal{L}_{d}$ with

$$u_{n}(\mu)-F_{1}(\iota_{d}(\mu))\le u_{n}(\mu_{n})-F_{1}(\iota_{d}(\mu_{n}))\qquad(\mu\in\mathcal{L}_{d}).$$

In particular $u_{n}-F_{1}\circ\iota_{d}$ has a local maximum at $\mu_{n}$ relative to $\mathcal{L}_{d}$ (with radius $1$), and since $u_{n}$ is a viscosity subsolution and $F_{1}\in C^{1}(E_{d})$, Viscosity Sub- and Supersolutions of the Hamilton-Jacobi-Bellman Equation on Unitary Laws §subsolution gives

$$\rho\,u_{n}(\mu_{n})+T_{F_{1}}(\mu_{n})\le0\qquad(n\in\mathbb{N}).\tag{Sn}$$

Step 3 (Localisation). Put $a_{n}=u_{n}(\mu_{n})-F_{1}(\iota_{d}(\mu_{n}))$ and $e_{n}=w(\lambda_{n})-\tfrac1n-F_{1}(\iota_{d}(\lambda_{n}))$. By the choice of $\mu_{n}$ and of $u_{n}$, $a_{n}\ge u_{n}(\lambda_{n})-F_{1}(\iota_{d}(\lambda_{n}))>e_{n}$. Since $u_{n}(\mu_{n})\le w(\mu_{n})\le w^{*}(\mu_{n})$, (ST) gives $a_{n}\le\Gamma(\mu_{n})\le\Gamma_{0}$. The sequence $(1/n)_{n}$ converges to $0$ by The Archimedean Property of the Real Numbers, and $(F_{1}(\iota_{d}(\lambda_{n})))_{n}$ converges to $F_{1}(\iota_{d}(\lambda_{0}))$ by (C2); by claims 1 and 3 of Arithmetic of Limits of Real Sequences, $(e_{n})_{n}$ converges to $w^{*}(\lambda_{0})-F_{1}(\iota_{d}(\lambda_{0}))=\Gamma_{0}$. By claim 2 of Order Properties of Limits of Real Sequences (with the constant sequence $\Gamma_{0}$ as upper sequence), $(a_{n})_{n}$ converges to $\Gamma_{0}$.

$(\mu_{n})_{n}$ converges to $\lambda_{0}$: let $\varepsilon>0$ and $s=\min\{\varepsilon,1\}$. Choose $N\in\mathbb{N}$ with $a_{n}>\Gamma_{0}-s^{2}$ for $n\ge N$. For such $n$, $d_{\mathcal{L}}(\mu_{n},\lambda_{0})\ge s$ would give $a_{n}\le\Gamma(\mu_{n})\le\Gamma_{0}-s^{2}$ by (ST), which is impossible; so $d_{\mathcal{L}}(\mu_{n},\lambda_{0})<s\le\varepsilon$ (Convergent Sequence in a Metric Space).

Consequently $(F_{1}(\iota_{d}(\mu_{n})))_{n}$ converges to $F_{1}(\iota_{d}(\lambda_{0}))$ by (C2), and $u_{n}(\mu_{n})=a_{n}+F_{1}(\iota_{d}(\mu_{n}))$ converges to $\Gamma_{0}+F_{1}(\iota_{d}(\lambda_{0}))=w^{*}(\lambda_{0})$ by claim 1 of Arithmetic of Limits of Real Sequences.

Step 4 (Passage to the limit). By (C2) with $G=F_{1}$, $(T_{F_{1}}(\mu_{n}))_{n}$ converges to $T_{F_{1}}(\lambda_{0})$; with Step 3 and claims 1 and 3 of Arithmetic of Limits of Real Sequences, $(\rho\,u_{n}(\mu_{n})+T_{F_{1}}(\mu_{n}))_{n}$ converges to $\rho\,w^{*}(\lambda_{0})+T_{F_{1}}(\lambda_{0})$. By (Sn) and claim 1 of Order Properties of Limits of Real Sequences (compared with the constant sequence $0$), $\rho\,w^{*}(\lambda_{0})+T_{F_{1}}(\lambda_{0})\le0$, and by (G0) this is

$$\rho\,w^{*}(\lambda_{0})+H\bigl(\lambda_{0},DF(\iota_{d}(\lambda_{0}))\bigr)-\frac{\beta^{2}}{2}\bigl\langle DF(\iota_{d}(\lambda_{0})),\widehat{\Theta}\lambda_{0}\bigr\rangle_{d}\le f(\lambda_{0}).$$

Together with the boundedness and upper semicontinuity of $w^{*}$ shown above, Viscosity Sub- and Supersolutions of the Hamilton-Jacobi-Bellman Equation on Unitary Laws §subsolution shows that $w^{*}$ is a viscosity subsolution of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation with Hamiltonian $H$.