Proof of Comparison Principle for Plan-Jet Viscosity Solutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws

Each result cited is universally quantified over the data in its own statement. By Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing and Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space, the $L^{2}$ $d$-tuples of a tracial W*-probability space $(H,M,\Omega)$ lie in the complex Hilbert space $H^{d}$, with its sums, real multiples, norm $\lVert\cdot\rVert_{2}$ and inner product $\langle\cdot,\cdot\rangle_{2}$, which is real, symmetric and real bilinear on $L^{2}$ $d$-tuples (conditions 1, 2 and 3 of Complex Inner Product Space and realness). Hence $\lVert P+Q\rVert_{2}^{2}=\lVert P\rVert_{2}^{2}+2\langle P,Q\rangle_{2}+\lVert Q\rVert_{2}^{2}$, $|\langle P,Q\rangle_{2}|\le\lVert P\rVert_{2}\lVert Q\rVert_{2}$ by Cauchy-Schwarz Inequality in a Complex Inner Product Space, and the triangle inequality, the reverse triangle inequality and homogeneity of $\lVert\cdot\rVert_{2}$ hold by claim 2 of The Induced Norm is a Norm, and Induces a Metric. If $\mathrm{law}(X,P)=\mathrm{law}(X',P')$ for $L^{2}$ $d$-tuples of possibly different spaces, then $\lVert X\rVert_{2}=\lVert X'\rVert_{2}$ and $\lVert P\rVert_{2}=\lVert P'\rVert_{2}$, by Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling, Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments and claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; we call this (N).

Step 0 (a common form). In case 1 let $\mathcal{H}_{0}=\mathcal{H}$, $\kappa=0$ and let $L$ be the constant of Hamiltonians on Phase-Space Noncommutative Laws that are Lipschitz in the Momentum with Linear Growth §lipschitz. In case 2 let $\mathcal{H}_{0}$, $L$ and $C$ be as in Quadratic Hamiltonians with a Convex Remainder Lipschitz in the Momentum on Phase-Space Noncommutative Laws and $\kappa=1$. In both cases $\mathcal{H}_{0}$ is Lipschitz in the momentum with linear growth with constant $L$ (Quadratic Hamiltonians with a Convex Remainder Lipschitz in the Momentum on Phase-Space Noncommutative Laws §lipschitz in case 2), and $\mathcal{H}_{M}(X,P)=\frac{\kappa}{2}\lVert P\rVert_{2}^{2}+\mathcal{H}_{0,M}(X,P)$ for all $L^{2}$ $d$-tuples $X,P$ (in case 2 by Quadratic Hamiltonians with a Convex Remainder Lipschitz in the Momentum on Phase-Space Noncommutative Laws §decomposition, since $\widehat{M}(\mathrm{pr}^{2}_{\#}\mathrm{law}(X,P))=\widehat{M}(\mathrm{law}(P))=\lVert P\rVert_{2}^{2}$ by Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling and Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments). Expanding $\lVert P+Q\rVert_{2}^{2}$, for all $L^{2}$ $d$-tuples $X,P,Q$ of one space,

Step 1 (set-up and the scaled subsolution). Let $B\ge0$ be real with $|u|\le B$ and $|v|\le B$ on $\Sigma^{2}_{d}$ (Bounded Real-Valued Function on a Set). Suppose, for a contradiction, that $\theta_{0}=u(\mu_{0})-v(\mu_{0})>0$ for some $\mu_{0}\in\Sigma^{2}_{d}$, and put $\eta'=\rho\theta_{0}/32$. In case 1 put $\theta=1$, $c=0$ and $E=0$. In case 2 put $C^{+}=\max\{C,0\}$, $s=\min\{\theta_{0}/(2B+2),\;\eta'/(2C^{+}+8),\;1/2\}$, $\theta=1-s$, $c=(1-\theta)/(2\theta)$ and $E=(1-\theta)C^{+}$; then $0<\theta<1$, $(1-\theta)B\le\theta_{0}/2$, $c\le s$ (as $\theta\ge1/2$) and $E+4c\le s(C^{+}+4)\le\eta'/2$. Let $w=\theta u$. Then $|w|\le B$, $w$ is upper semicontinuous by claim 2 of Sums and Nonnegative Multiples of Semicontinuous Functions, and $\theta_{1}=w(\mu_{0})-v(\mu_{0})=\theta_{0}-(1-\theta)u(\mu_{0})\ge\theta_{0}/2>0$. Moreover:

(S) for every real $\delta\ge0$, $\mu\in\Sigma^{2}_{d}$, $\pi\in J^{+}_{\delta}w(\mu)$ and real $\eta>0$ there are a tracial W*-probability space $(H_{1},N_{1},\Psi_{1})$ and $L^{2}$ $d$-tuples $X,P,Q$ of it with $\mathrm{law}(X,P)=\pi$, $\lVert Q\rVert_{2}\le\delta$ and $\rho\,w(\mu)+\mathcal{H}_{N_{1}}(X,P+Q)+c\lVert P+Q\rVert_{2}^{2}\le E+\eta$.

In case 1 this is Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions §sub; in case 2 it is Scaling a Plan-Jet Viscosity Subsolution of an Equation with a Quadratic Hamiltonian Gives a Strict Subsolution §scaling, since $(1-\theta)C\le E$.

Step 2 (order of choice). Let $G$ and, for reals $\varepsilon',\beta'>0$, the doubled functional $\Phi_{\varepsilon'}$ be as in Plan Jets at an Ekeland Point of the Doubled Difference with Logarithmic Confinement on Square-Integrable Noncommutative Laws, with $w$ in place of $u$ and with $\beta'=\beta$ fixed in (i); $G\ge0$ since $\log(1+s')\ge1-(1+s')^{-1}\ge0$ for $s'\ge0$ by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §log. The constants are chosen in the following order.

(i) $\beta=\min\{1,\;\theta_{1}/(8(1+G(\mu_{0}))),\;\eta'/(6L+1)\}$, and in case 2 additionally $\beta\le(c\eta'/8)^{1/2}$ (replace $\beta$ by the minimum). Then $0<\beta\le1$, $2\beta G(\mu_{0})\le\theta_{1}/4$, $6L\beta+\beta^{2}\le(6L+1)\beta\le\eta'$, and in case 2, $8\beta^{2}/c\le\eta'$.

(ii) $R\ge0$ with $R^{2}=\exp(2B/\beta)-1$, which is $\ge0$ by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §exp and exists by Existence and Uniqueness of the Nonnegative Square Root. By The Structure Condition for a Hamiltonian on Phase-Space Noncommutative Laws §structure for $R$ and $\eta'$, fix $r_{s}>0$.

(iii) For $\varepsilon'>0$, $\Phi_{\varepsilon'}\le2B$ on $\Sigma^{2}_{2d}$, since $|w|,|v|\le B$, $\mathcal{I}\ge0$ (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 §cost) and $G\ge0$; so $\mathsf{M}(\varepsilon')=\sup\Phi_{\varepsilon'}$ exists (Approximation Property of the Supremum and the Infimum in $\mathbb{R}$). If $\varepsilon''\le\varepsilon'$ then $\Phi_{\varepsilon''}\le\Phi_{\varepsilon'}$ pointwise, since $\mathcal{I}\ge0$ (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 §cost); so $\mathsf{M}(\varepsilon'')\le\mathsf{M}(\varepsilon')$. With $\gamma_{0}=\mathrm{diag}_{\#}\mu_{0}$, 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 §diagonal and Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §couplings give $\Phi_{\varepsilon'}(\gamma_{0})=\theta_{1}-2\beta G(\mu_{0})\ge\frac34\theta_{1}$ for every $\varepsilon'$. Put $m_{n}=\mathsf{M}(2^{-n})$ for $n\in\mathbb{N}$: $(m_{n})$ is nonincreasing and bounded below by $\frac34\theta_{1}$, so $m_{*}=\inf_{n}m_{n}$ exists, and by claim 4 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ there is $N$ with $m_{N}<m_{*}+r_{s}/16$; hence $m_{n}-m_{n+1}\le m_{N}-m_{*}<r_{s}/16$ for every $n\ge N$. By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric fix $n\ge N$ with $2^{-(n+1)}\le r_{s}^{2}/(16(B+1))$, and put $\varepsilon=2^{-(n+1)}$, $\Phi=\Phi_{\varepsilon}$ and $\Phi'=\Phi_{2\varepsilon}$; thus $\sup\Phi'-\sup\Phi=m_{n}-m_{n+1}<r_{s}/16$.

(iv) $\delta=\min\{\beta,\;\eta'/(2L(1+R)+3)\}$; then $0<\delta\le\beta\le1$ and $2L(1+R)\delta+2\delta+\delta^{2}\le\eta'$.

(v) $\eta_{E}=\min\{\theta_{1}/4,\;r_{s}/16\}$ and $\kappa_{E}=\eta_{E}/\delta$.

Step 3 (Ekeland point). $(\Sigma^{2}_{2d},\widehat{W}_{2})$ is a complete metric space by Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §laws, The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §metric and The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §complete. $\Phi$ is upper semicontinuous: the maps $\mathrm{pr}^{1}_{\#},\mathrm{pr}^{2}_{\#},D_{\#},\widehat{M}$ are continuous 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 §lipschitz; $\mathcal{I}=\widehat{M}\circ D_{\#}$ (Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §cost) and $G\circ\mathrm{pr}^{i}_{\#}$ are continuous by claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map, $s'\mapsto\log(1+s')$ being continuous on $[0,\infty)$ as the composite of $s'\mapsto1+s'$ with the smooth map $\log$ of The Natural Logarithm (continuous by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous); $w\circ\mathrm{pr}^{1}_{\#}$ is upper and $v\circ\mathrm{pr}^{2}_{\#}$ lower semicontinuous by claims 1 and 2 of Semicontinuity and Continuity Under Composition with a Continuous Map; a continuous function and the negative of a lower semicontinuous function are upper semicontinuous directly from Continuous Map Between Metric Spaces, Upper Semicontinuous Function on a Subset of a Metric Space and Lower Semicontinuous Function on a Subset of a Metric Space; and sums and nonnegative multiples of upper semicontinuous functions are upper semicontinuous by claims 1 and 2 of Sums and Nonnegative Multiples of Semicontinuous Functions. $\Phi\le2B$ by (iii).

By claim 3 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ there is $\gamma_{s}$ with $\Phi(\gamma_{s})>\sup\Phi-\eta_{E}$. By Ekeland's Variational Principle for Upper Semicontinuous Functions on a Complete Metric Space with $x_{0}=\gamma_{s}$, $\eta_{E}$ and $\kappa_{E}$, there is $\hat{\gamma}$ with $\Phi(\hat{\gamma})\ge\Phi(\gamma_{s})\ge\sup\Phi-\eta_{E}$ (Ekeland's Variational Principle for Upper Semicontinuous Functions on a Complete Metric Space §value) and $\Phi(\gamma)-\delta\,\widehat{W}_{2}(\gamma,\hat{\gamma})\le\Phi(\hat{\gamma})$ for every $\gamma$ (Ekeland's Variational Principle for Upper Semicontinuous Functions on a Complete Metric Space §perturbed for $\gamma\ne\hat{\gamma}$, trivially for $\gamma=\hat{\gamma}$). In particular $\Phi(\hat{\gamma})\ge\Phi(\gamma_{0})-\eta_{E}\ge\frac34\theta_{1}-\frac14\theta_{1}=\frac12\theta_{1}$.

Step 4 (realisation and bounds). By Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling there are $(H,M,\Omega)$ and $L^{2}$ $d$-tuples $\hat{X},\hat{Y}$ with $\mathrm{law}(\hat{X},\hat{Y})=\hat{\gamma}$; let $\lambda=\mathrm{law}(\hat{X})$, $\nu=\mathrm{law}(\hat{Y})$, $D=\lVert\hat{X}-\hat{Y}\rVert_{2}$, and $p,q,q'$ as in Plan Jets at an Ekeland Point of the Doubled Difference with Logarithmic Confinement on Square-Integrable Noncommutative Laws. By the same citation, $\mathrm{pr}^{1}_{\#}\hat{\gamma}=\lambda$, $\mathrm{pr}^{2}_{\#}\hat{\gamma}=\nu$, $\mathcal{I}(\hat{\gamma})=D^{2}$, and $\widehat{M}(\lambda)=\lVert\hat{X}\rVert_{2}^{2}$, $\widehat{M}(\nu)=\lVert\hat{Y}\rVert_{2}^{2}$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments; so

all added terms being $\ge0$. (a) $w(\lambda)-v(\nu)\ge\frac12\theta_{1}\ge\frac14\theta_{0}$. (b) $\beta\log(1+\lVert\hat{X}\rVert_{2}^{2})\le2B$, so, $\exp$ being strictly increasing (claim 4 of Basic Properties of the Exponential Function) with $\exp(\log t)=t$ (The Natural Logarithm), $\lVert\hat{X}\rVert_{2}^{2}\le R^{2}$ and $\lVert\hat{X}\rVert_{2}\le R$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; likewise $\lVert\hat{Y}\rVert_{2}\le R$. (c) $\frac{1}{2\varepsilon}D^{2}\le2B$, so $D^{2}\le4B\varepsilon\le r_{s}^{2}/4$ and $D\le r_{s}/2$; and by Plan Jets at an Ekeland Point of the Doubled Difference with Logarithmic Confinement on Square-Integrable Noncommutative Laws §penalisation, whose constant is here taken to be $\eta_{E}$ (Step 3), $\frac{1}{4\varepsilon}D^{2}\le m_{n}-m_{n+1}+\eta_{E}<r_{s}/8$. Hence, with $\alpha=1/\varepsilon$, $\alpha D^{2}+D<r_{s}$, and $p=\alpha(\hat{X}-\hat{Y})$. (d) By Plan Jets at an Ekeland Point of the Doubled Difference with Logarithmic Confinement on Square-Integrable Noncommutative Laws §confinement, $\lVert q\rVert_{2},\lVert q'\rVert_{2}\le\beta$, $(1+\lVert\hat{X}\rVert_{2})\lVert q\rVert_{2}\le3\beta$ and $(1+\lVert\hat{Y}\rVert_{2})\lVert q'\rVert_{2}\le3\beta$.

Step 5 (the subsolution side). Let $\pi_{u}=\mathrm{law}(\hat{X},p+q)$; by Plan Jets at an Ekeland Point of the Doubled Difference with Logarithmic Confinement on Square-Integrable Noncommutative Laws §superjet (with $w$), $\pi_{u}\in J^{+}_{\delta}w(\lambda)$. By (S) with $\eta=\eta'$ there are $(H_{1},N_{1},\Psi_{1})$ and $X_{1},P_{1},Q_{1}$ with $\mathrm{law}(X_{1},P_{1})=\pi_{u}$, $\lVert Q_{1}\rVert_{2}\le\delta$ and $\rho\,w(\lambda)+\mathcal{H}_{N_{1}}(X_{1},P_{1}+Q_{1})+c\lVert P_{1}+Q_{1}\rVert_{2}^{2}\le E+\eta'$. By (N), $\lVert X_{1}\rVert_{2}=\lVert\hat{X}\rVert_{2}\le R$ and $\lVert P_{1}\rVert_{2}=\lVert p+q\rVert_{2}$, so $\lVert p\rVert_{2}-\beta\le\lVert P_{1}\rVert_{2}\le\lVert p\rVert_{2}+\beta$ and $\lVert P_{1}+Q_{1}\rVert_{2}\ge\lVert p\rVert_{2}-(\beta+\delta)$. For reals $a,s',t\ge0$ with $a\ge s'-t$ one has $a^{2}\ge\frac12s'^{2}-t^{2}$: if $s'\le t$ the right side is $\le0$; otherwise $a^{2}\ge(s'-t)^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, and $(s'-t)^{2}-\frac12s'^{2}+t^{2}=\frac12(s'-2t)^{2}\ge0$. Hence $c\lVert P_{1}+Q_{1}\rVert_{2}^{2}\ge\frac{c}{2}\lVert p\rVert_{2}^{2}-c(\beta+\delta)^{2}\ge\frac{c}{2}\lVert p\rVert_{2}^{2}-4c$. By (M$-$) with $X_{1},P_{1},Q_{1}$ and $\mathcal{H}_{N_{1}}(X_{1},P_{1})=\mathcal{H}(\pi_{u})$ (Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts),

Step 6 (the supersolution side). Let $\pi_{v}=\mathrm{law}(\hat{Y},p-q')$; by Plan Jets at an Ekeland Point of the Doubled Difference with Logarithmic Confinement on Square-Integrable Noncommutative Laws §subjet, $\pi_{v}\in J^{-}_{\delta}v(\nu)$. By Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions §super with $\eta'$ there are $(H_{2},N_{2},\Psi_{2})$ and $Y_{2},P_{2},Q_{2}$ with $\mathrm{law}(Y_{2},P_{2})=\pi_{v}$, $\lVert Q_{2}\rVert_{2}\le\delta$ and $\rho\,v(\nu)+\mathcal{H}_{N_{2}}(Y_{2},P_{2}+Q_{2})\ge-\eta'$. By (N), $\lVert Y_{2}\rVert_{2}\le R$ and $\lVert P_{2}\rVert_{2}=\lVert p-q'\rVert_{2}\le\lVert p\rVert_{2}+\beta$, so by (M+),

Step 7 (three brackets). In $(H,M,\Omega)$,

By (M+) with $P=p$, $Q=-q'$ and Step 4(d), the first bracket is $\le\kappa\beta\lVert p\rVert_{2}+\beta^{2}+3L\beta$. By (M$-$) with $P=p$, $Q=q$, the third is $\le\kappa\beta\lVert p\rVert_{2}+3L\beta$. By Step 4(b),(c) and the choice of $r_{s}$ in (ii), the second is $<\eta'$. With (i), $\mathcal{H}(\pi_{v})-\mathcal{H}(\pi_{u})<2\kappa\beta\lVert p\rVert_{2}+2\eta'$.

Step 8 (contradiction). Adding (U), (V) and Step 7, and using $\kappa\le1$, $\beta\le1$,

By (iv), $2\delta+\delta^{2}+2L(1+R)\delta\le\eta'$. In case 1, $\kappa=c=E=0$, so the right side is $\le5\eta'$. In case 2, $E+4c\le\eta'$, and for every real $s'$, $-\frac{c}{2}s'^{2}+2(\beta+\delta)s'\le2(\beta+\delta)^{2}/c\le8\beta^{2}/c\le\eta'$ (complete the square; $\delta\le\beta$; (i)); so the right side is $\le7\eta'$. In both cases $\rho(w(\lambda)-v(\nu))<7\eta'=\frac{7}{32}\rho\theta_{0}$, whereas Step 4(a) and $\rho>0$ give $\rho(w(\lambda)-v(\nu))\ge\frac14\rho\theta_{0}=\frac{8}{32}\rho\theta_{0}$. This contradiction shows $u(\mu)\le v(\mu)$ for every $\mu\in\Sigma^{2}_{d}$, in both cases.