Entropy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi-Bellman Equation 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, consider the equation $(\mathrm{E})$ of The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space §laplacian-form and its score form $(\mathrm{E}_{\xi})$ of The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space §score-form, and let $u:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$. For an intrinsic test function $\varphi$ on $\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$ and $\mu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ the gradient $\nabla\varphi(\mu)$ is defined, $\mu$ being absolutely continuous by The Entropy of Measures on the Torus: Nonnegativity, Absolute Continuity and Closed Sublevel Sets §nonnegative.

1. (Subsolution)¶ $u$ is an entropy-penalised viscosity subsolution of $(\mathrm{E})$ if there are a real number $\delta_{0}>0$ and a nondecreasing function $c:[0,\infty)\to[0,\infty)$ with the following property. For every intrinsic test function $\varphi$ on $\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$, every real $\delta$ with $0<\delta\le\delta_{0}$, and every $\mu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ such that

$$u(\nu)-\varphi(\nu)-\delta\,\mathrm{Ent}(\nu)<u(\mu)-\varphi(\mu)-\delta\,\mathrm{Ent}(\mu)\qquad\text{for every }\nu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})\text{ with }\nu\ne\mu,$$

one has $\mu\in\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$ and

$$\rho\,u(\mu)+H\bigl(\mu,\nabla\varphi(\mu)\bigr)+\frac{\sigma^{2}}{2}\,\bigl\langle\nabla\varphi(\mu),\xi_{\mu}\bigr\rangle_{\mu}\le\delta\,c\bigl(\lVert\nabla\varphi(\mu)\rVert_{\mu}\bigr).$$

2. (Supersolution)¶ $u$ is an entropy-penalised viscosity supersolution of $(\mathrm{E})$ if there are a real number $\delta_{0}>0$ and a nondecreasing function $c:[0,\infty)\to[0,\infty)$ with the following property. For every intrinsic test function $\varphi$ on $\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$, every real $\delta$ with $0<\delta\le\delta_{0}$, and every $\mu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ such that

$$u(\nu)-\varphi(\nu)+\delta\,\mathrm{Ent}(\nu)>u(\mu)-\varphi(\mu)+\delta\,\mathrm{Ent}(\mu)\qquad\text{for every }\nu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})\text{ with }\nu\ne\mu,$$

one has $\mu\in\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$ and

$$\rho\,u(\mu)+H\bigl(\mu,\nabla\varphi(\mu)\bigr)+\frac{\sigma^{2}}{2}\,\bigl\langle\nabla\varphi(\mu),\xi_{\mu}\bigr\rangle_{\mu}\ge-\delta\,c\bigl(\lVert\nabla\varphi(\mu)\rVert_{\mu}\bigr).$$

3. (Solution)¶ $u$ is an entropy-penalised viscosity solution of $(\mathrm{E})$ if it is both an entropy-penalised viscosity subsolution and an entropy-penalised viscosity supersolution of $(\mathrm{E})$.