The Hamilton-Jacobi Equation for Controlled Langevin Dynamics in a Potential on Euclidean Space

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number, let $V:\mathbb{R}^{n}\to\mathbb{R}$ be of class $C^{2}$ 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 and let $g:\mathbb{R}^{n}\to\mathbb{R}$.

1. (The operator)¶ The Langevin Hamilton-Jacobi operator with potential $V$, discount $\lambda$, control cost $\theta$, noise intensity $\kappa$ and running cost $g$ is the penalty-drift Hamilton-Jacobi operator on $D=\mathbb{R}^{n}$ with potential $V$ and these coefficients:

2. (The equation)¶ The Langevin Hamilton-Jacobi equation is the penalty-drift Hamilton-Jacobi equation for these data,