Proof of Scaling a Plan-Jet Viscosity Subsolution of an Equation with a Quadratic Hamiltonian Gives a Strict Subsolution

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 lie in the complex Hilbert space $H^{d}$ with its sums, real multiples, norm and inner product $\langle\cdot,\cdot\rangle_{2}$, real and real bilinear on $L^{2}$ $d$-tuples (conditions 1, 2 and 3 of Complex Inner Product Space and realness), and $\lVert tZ\rVert_{2}=|t|\lVert Z\rVert_{2}$ by claim 2 of The Induced Norm is a Norm, and Induces a Metric. Let $\mathcal{H}_{0}$ and $L$ be as in Quadratic Hamiltonians with a Convex Remainder Lipschitz in the Momentum on Phase-Space Noncommutative Laws, with the given $C$. By Quadratic Hamiltonians with a Convex Remainder Lipschitz in the Momentum on Phase-Space Noncommutative Laws §decomposition, 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, $\mathcal{H}_{M}(X,V)=\frac12\lVert V\rVert_{2}^{2}+\mathcal{H}_{0,M}(X,V)$ for all $L^{2}$ $d$-tuples $X,V$ of any tracial W*-probability space $(H,M,\Omega)$.

Let $\delta\ge0$, $\mu$, $\pi\in J^{+}_{\delta}w(\mu)$ and $\eta>0$ be given. Let $S$ be the affine datum from $2d$ to $2d$ variables with $S(X,P)=(X,\theta P)$ for all $L^{2}$ $d$-tuples $X,P$ (diagonal entries $1$ on the first $d$ coordinates and $\theta$ on the last $d$, all other entries and the translation part $0$; Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations).

Step 1 (a rescaled plan). 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 $L^{2}$ $d$-tuples $X_{0},P_{0}$ of some tracial W*-probability space with $\mathrm{law}(X_{0},P_{0})=\pi$; put $\pi'=\mathrm{law}(X_{0},\theta^{-1}P_{0})$. If $X,P'$ are $L^{2}$ $d$-tuples of any tracial W*-probability space with $\mathrm{law}(X,P')=\pi'$, then by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward (twice)

By Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling, $\mathrm{pr}^{1}_{\#}\pi'=\mathrm{law}(X_{0})=\mathrm{pr}^{1}_{\#}\pi=\mu$, the last equality because $\pi$ is a plan at $\mu$ (Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §superjet); so $\pi'$ is a plan at $\mu$.

Step 2 ($\pi'$ is a plan superjet of $u$ with slack $\delta/\theta$). Let $\eta_{1}>0$. By Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super for $w$ with $\theta\eta_{1}$ in place of $\eta$, fix $r>0$. Let $X,P',X'$ be $L^{2}$ $d$-tuples of a tracial W*-probability space $(K,N,\Psi)$ with $\mathrm{law}(X,P')=\pi'$ and $\lVert X'-X\rVert_{2}<r$. By (R), $\mathrm{law}(X,\theta P')=\pi$, so

Dividing by $\theta>0$ shows that $\pi'$ is a plan superdifferential of $u$ at $\mu$ with slack $\delta/\theta$ (Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super), so $\pi'\in J^{+}_{\delta/\theta}u(\mu)$ by Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §superjet.

Step 3 (the subsolution inequality). By Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions §sub applied to $u$ with slack $\delta/\theta$, $\mu$, $\pi'$ and $\eta/\theta$, there are $(H,M,\Omega)$ and $L^{2}$ $d$-tuples $X,P',Q'$ of it with $\mathrm{law}(X,P')=\pi'$, $\lVert Q'\rVert_{2}\le\delta/\theta$ and $\rho\,u(\mu)+\mathcal{H}_{M}(X,P'+Q')\le\eta/\theta$. Put $P=\theta P'$, $Q=\theta Q'$ and $Z=P+Q$. Then $\mathrm{law}(X,P)=\pi$ by (R), $\lVert Q\rVert_{2}=\theta\lVert Q'\rVert_{2}\le\delta$, and $P'+Q'=\theta^{-1}Z$. Multiplying by $\theta>0$,

Step 4 (convexity). By Quadratic Hamiltonians with a Convex Remainder Lipschitz in the Momentum on Phase-Space Noncommutative Laws §convex with $t=\theta$, the momenta $\theta^{-1}Z$ and $0$, and $\theta(\theta^{-1}Z)+(1-\theta)0=Z$, and then Quadratic Hamiltonians with a Convex Remainder Lipschitz in the Momentum on Phase-Space Noncommutative Laws §bound with $1-\theta>0$,

Since $\frac{1}{2\theta}=\frac12+\frac{1-\theta}{2\theta}$,

Combining with Step 3 gives $\rho\,w(\mu)+\mathcal{H}_{M}(X,P+Q)+\frac{1-\theta}{2\theta}\lVert P+Q\rVert_{2}^{2}\le(1-\theta)C+\eta$.