Well-Posedness of the Langevin Hamilton-Jacobi Equation on Euclidean Space under a Dissipation Inequality

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number, let $V$ be a penalty on $\mathbb{R}^{n}$, which is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, with the notation of The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space for traces, squared norms and halves, let $\lambda\in\mathbb{R}$ be positive, let $\theta,\kappa\in\mathbb{R}$ be nonnegative, let $g:\mathbb{R}^{n}\to\mathbb{R}$ be continuous into the real line, let $M\in\mathbb{R}$ satisfy $|g(x)|\le M$ for every $x\in\mathbb{R}^{n}$, and let $F$ be the Langevin Hamilton-Jacobi operator with potential $V$, discount $\lambda$, control cost $\theta$, noise intensity $\kappa$ and running cost $g$. Assume the dissipation inequality:¶ there are $\varepsilon,C\in\mathbb{R}$ with $0<\varepsilon\le1$ such that

Then the following hold.

1. (Comparison)¶ If $u:\mathbb{R}^{n}\to\mathbb{R}$ is a viscosity subsolution of $F$ on $\mathbb{R}^{n}$ with $V$-subordinate growth from above and $v:\mathbb{R}^{n}\to\mathbb{R}$ is a viscosity supersolution of $F$ on $\mathbb{R}^{n}$ with $V$-subordinate growth from below, then $u(x)\le v(x)$ for every $x\in\mathbb{R}^{n}$.

2. (Existence and uniqueness)¶ There is exactly one function $u:\mathbb{R}^{n}\to\mathbb{R}$ that is both a viscosity subsolution and a viscosity supersolution of $F$ on $\mathbb{R}^{n}$ and has $V$-subordinate growth from above and from below. It is continuous on $\mathbb{R}^{n}$ and satisfies $-\lambda^{-1}M\le u(x)\le\lambda^{-1}M$ for every $x\in\mathbb{R}^{n}$.