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

This setting fixes the standing notation for the discounted Hamilton-Jacobi-Bellman equation with idiosyncratic noise on the torus Wasserstein space. It introduces no new concept and asserts nothing beyond the identifications recorded below, each justified by the reference attached to it.

1. (Calculus)¶ The notation of Heat Smoothing, Entropy, Fisher Information and Laplacian Test Functions on the Torus Wasserstein Space: Standing Notation is in force.

2. (Parameters)¶ $\rho$ and $\sigma$ denote real numbers with $\rho>0$ and $\sigma>0$: the discount rate and the noise intensity.

3. (Hamiltonian)¶ $H$ denotes a map, the Hamiltonian, that assigns a real number $H(\mu,p)$ to every $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and every $p\in T_{\mu}$. For a subset $\mathcal{K}$ of $\mathcal{P}(\mathbb{T}^{d})$, an intrinsic test function $\varphi$ on $\mathcal{K}$ and $\mu\in\mathcal{K}$, one has $\nabla\varphi(\mu)\in T_{\mu}$ by Intrinsic Test Functions on the Torus Wasserstein Space §differentiability, so $H(\mu,\nabla\varphi(\mu))$ is defined.

4. (Background)¶ The following results are in force by reference: Heat Regularisation of an Intrinsic Test Function: a Laplacian Test Function whose Laplacian is Minus the Pairing of the Gradient with the Score and Heat Regularisation of the Entropy: a Laplacian Test Function whose Laplacian is Minus the Fisher Information, together with those of Heat Smoothing, Entropy, Fisher Information and Laplacian Test Functions on the Torus Wasserstein Space: Standing Notation §background.