The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space

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, let $P:D\to\mathbb{R}$ be of class $C^{2}$ on $D$, let $\lambda\in\mathbb{R}$ be positive, let $\theta,\kappa\in\mathbb{R}$ be nonnegative and let $g:D\to\mathbb{R}$. We write $\operatorname{tr}$ for the trace, abbreviate $\lVert z\rVert^{2}=\lVert z\rVert\lVert z\rVert$, and write $\tfrac{a}{2}$ for the product of $a\in\mathbb{R}$ with the multiplicative inverse of $2=1+1$.

1. (The operator)¶ The penalty-drift Hamilton-Jacobi operator with potential $P$, discount $\lambda$, control cost $\theta$, noise intensity $\kappa$ and running cost $g$ is the second-order equation operator on $D$

2. (The equation)¶ The penalty-drift Hamilton-Jacobi equation is

that is, $F(x,u(x),Du(x),D^{2}u(x))=0$ for $x\in D$. Its classical sub- and supersolutions are those of Classical Subsolution and Supersolution of a Second-Order Equation and its viscosity sub- and supersolutions those of Viscosity Subsolution and Supersolution of a Second-Order Equation, for the operator $F$ on $D$.