Well-Posedness of the Langevin Hamilton-Jacobi Equation with Common Noise and a Density Cost: Existence and Uniqueness of a Bounded Viscosity Solution

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation. Let $V$ be a confining potential on $\mathbb{R}^{d}$, let $\lambda_{0},\sigma\in\mathbb{R}$ be positive, let $\theta\in\mathbb{R}$ satisfy $0<\theta\le1$, let $\kappa,L\in\mathbb{R}$ be nonnegative, let $\Phi$ be a convex Lipschitz integrand with constant $L$, and let $g:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be uniformly continuous for $W_{2}$ and the metric of The Absolute Value Metric on the Real Line, with $b\in\mathbb{R}$ such that $|g(\nu)|\le b$ for every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Langevin free-energy pair with potential $V$ and noise intensity $\sigma$. Viscosity solutions, subsolutions and supersolutions of the Langevin Hamilton-Jacobi equation with common noise and density cost with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, common-noise intensity $\kappa$, control cost $\theta$, running cost $g$ and integrand $\Phi$ are those of that clause, functions on $\mathcal{D}$; $\lambda_{0}^{-1}$ is the multiplicative inverse of $\lambda_{0}$, and uniform continuity on a subset of $\mathcal{D}$ refers to $W_{2}$ restricted to that subset. The letter $\sigma$ denotes the noise intensity; the swap map written $\sigma$ in Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §swap is not used.

1. (Comparison)¶ Let $u,v:\mathcal{D}\to\mathbb{R}$ be a viscosity subsolution bounded above and a viscosity supersolution bounded below of that equation. Then $u(\mu)\le v(\mu)$ for every $\mu\in\mathcal{D}$.

2. (Existence)¶ There is a viscosity solution $u:\mathcal{D}\to\mathbb{R}$ of that equation with $-\lambda_{0}^{-1}b\le u(\mu)\le\lambda_{0}^{-1}(b+L)$ for every $\mu\in\mathcal{D}$.

3. (Uniqueness and continuity)¶ Any two bounded viscosity solutions of that equation are equal, and a bounded viscosity solution is uniformly continuous on $\{\mu\in\mathcal{D}:\mathcal{E}(\mu)\le c\}$ for every $c\in\mathbb{R}$.