Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number, let $D\subseteq\mathbb{R}^{n}$ be open and nonempty, let $P$ be a penalty on $D$, and for $R\in\mathbb{R}$ let $D_{R}=\{x\in D:P(x)<R\}$. 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 $g:D\to\mathbb{R}$ be continuous into the real line, let $M\in\mathbb{R}$ satisfy $|g(x)|\le M$ for every $x\in D$, and let $F$ be the penalty-drift Hamilton-Jacobi operator with potential $P$, discount $\lambda$, control cost $\theta$, noise intensity $\kappa$ and running cost $g$. Assume the following two conditions.

1. (One-sided drift)¶ For every $R\in\mathbb{R}$ there is a nonnegative $c_{R}\in\mathbb{R}$ with

2. (Dissipation)¶ There are $\varepsilon,C\in\mathbb{R}$ with $0<\varepsilon\le1$ such that

Then the following hold.

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

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