Lipschitz Dependence of the Bounded Viscosity Solution of the Langevin Hamilton-Jacobi Equation with a Density Cost on the Running Cost

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 $L\in\mathbb{R}$ be nonnegative, let $p\in\mathbb{N}$ and let $\Gamma\in\mathcal{M}_{p\times d}(\mathbb{R})$ be a real $p\times d$ matrix, and let $\Phi$ be a convex Lipschitz integrand with constant $L$. Let $g,g':\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be uniformly continuous for $W_{2}$ and the metric of The Absolute Value Metric on the Real Line, with $b,b'\in\mathbb{R}$ such that $|g(\nu)|\le b$ and $|g'(\nu)|\le b'$ for every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, where $|\cdot|$ is the absolute value; and let $k\in\mathbb{R}$ satisfy

Let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Langevin free-energy pair with potential $V$ and noise intensity $\sigma$. Viscosity solutions of the Langevin Hamilton-Jacobi equation with common noise and density cost, with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, common-noise matrix $\Gamma$, control cost $\theta$, integrand $\Phi$ and a running cost, are those of that clause, functions on $\mathcal{D}$; $\lambda_{0}^{-1}$ is the multiplicative inverse of $\lambda_{0}$. 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.

Let $u:\mathcal{D}\to\mathbb{R}$ be a viscosity solution of that equation with running cost $g$, and $u':\mathcal{D}\to\mathbb{R}$ a viscosity solution of that equation with running cost $g'$, and suppose both are bounded: there is $c\in\mathbb{R}$ with $|u(\mu)|\le c$ and $|u'(\mu)|\le c$ for every $\mu\in\mathcal{D}$.

(Lipschitz dependence on the running cost)¶ Then