The Linear-Quadratic Hamiltonian: Its Lift, the Structure Condition and Its Quadratic Structure

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $\rho$, $f$, the affine data $b_{\mu}=(A(\mu),c(\mu))$ ($\mu\in\Sigma^{2}_{d}$) and $\mathcal{H}^{\mathrm{LQ}}$ be as in The Linear-Quadratic Hamilton-Jacobi Equation with Law-Dependent Affine Drift on Square-Integrable Noncommutative Laws. Assume that there are reals $a\ge0$ and $L\ge0$ such that, for all $\mu,\nu\in\Sigma^{2}_{d}$ and $i,j\in[d]$,

and that $f$ is bounded and uniformly continuous on $\Sigma^{2}_{d}$, with the metrics of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics. Lifts, sums and the pairing are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts and Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing, and the $L^{2}$ norm is that of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples.

1. (Lift)¶ For every tracial W*-probability space $(H,M,\Omega)$ and all $L^{2}$ $d$-tuples $X,P$ of $(H,M,\Omega)$,

2. (Structure condition)¶ $\mathcal{H}^{\mathrm{LQ}}$ satisfies the structure condition.

3. (Quadratic structure)¶ $\mathcal{H}^{\mathrm{LQ}}$ is quadratic with a convex Lipschitz remainder.