The Linear-Quadratic Hamilton-Jacobi Equation with Law-Dependent Affine Drift on Square-Integrable Noncommutative Laws

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $\rho>0$ be real, let $f:\Sigma^{2}_{d}\to\mathbb{R}$, and for every $\mu\in\Sigma^{2}_{d}$ let $b_{\mu}=(A(\mu),c(\mu))$ be an affine datum from $d$ to $d$ variables. For $\mu\in\Sigma^{2}_{d}$ let $S_{\mu}=(B,e)$ be the affine datum from $2d$ to $2d$ variables with, for $i,j\in[d]$,

1. (Hamiltonian)¶ The linear-quadratic Hamiltonian with drift $b$ and running cost $f$ is $\mathcal{H}^{\mathrm{LQ}}:\Sigma^{2}_{2d}\to\mathbb{R}$,

with the push-forwards of Square-Integrable Noncommutative Laws: Standing Notation §affine and the real-valued second and quadratic moments of Square-Integrable Noncommutative Laws: Standing Notation §moments.

2. (Equation)¶ The linear-quadratic Hamilton--Jacobi equation with discount rate $\rho$, drift $b$ and running cost $f$ is the discounted stationary Hamilton--Jacobi equation with discount rate $\rho$ and Hamiltonian $\mathcal{H}^{\mathrm{LQ}}$, written formally, with the lifts of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts, the affine images $b_{\mathrm{law}(X)}X$ and the pairing of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing, as