The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space

In the setting of The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space: Standing Notation, with $\rho$, $\sigma$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space: Standing Notation §parameters and $H$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space: Standing Notation §hamiltonian.

1. (Laplacian form)¶ For a Laplacian test function $U$, the discounted Hamilton-Jacobi-Bellman equation with idiosyncratic noise is

$$(\mathrm{E})\qquad\rho\,U(\mu)+H\bigl(\mu,\nabla U(\mu)\bigr)-\frac{\sigma^{2}}{2}\,\mathcal{L}U(\mu)=0,\qquad\mu\in\mathcal{P}(\mathbb{T}^{d}),$$

with $\mathcal{L}U$ the Laplacian of $U$.

2. (Score form)¶ For an intrinsic test function $U$ on $\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$, the score form of $(\mathrm{E})$ is

$$(\mathrm{E}_{\xi})\qquad\rho\,U(\mu)+H\bigl(\mu,\nabla U(\mu)\bigr)+\frac{\sigma^{2}}{2}\,\bigl\langle\nabla U(\mu),\xi_{\mu}\bigr\rangle_{\mu}=0,\qquad\mu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})\cap\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d}),$$

with $\xi_{\mu}$ the score of $\mu$; members of $\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ are absolutely continuous by The Entropy of Measures on the Torus: Nonnegativity, Absolute Continuity and Closed Sublevel Sets §nonnegative, so $\nabla U(\mu)$ is defined there.