The Discounted Hamilton-Jacobi-Bellman Equation of Mean-Field Control with Idiosyncratic Noise on the Torus

In the setting of The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space: Standing Notation, with $\rho$ and $\sigma$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space: Standing Notation §parameters, let $(b,f)$ be a control datum.

¶The discounted Hamilton-Jacobi-Bellman equation of mean-field control with datum $(b,f)$ is the equation $(\mathrm{E})$ of The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space §laplacian-form with Hamiltonian $H=H_{b,f}$, the control Hamiltonian of $(b,f)$: for a Laplacian test function $U$,

$$(\mathrm{E}_{b,f})\qquad\rho\,U(\mu)+\int\Bigl(\tfrac12\lVert\nabla U(\mu)(x)\rVert^{2}-b(x,\mu)\cdot\nabla U(\mu)(x)-f(x,\mu)\Bigr)\,\mu(dx)-\frac{\sigma^{2}}{2}\,\mathcal{L}U(\mu)=0,\qquad\mu\in\mathcal{P}(\mathbb{T}^{d}).$$