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

Each result cited is universally quantified over the data in its own statement. We apply Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality with $D=\mathbb{R}^{n}$, which is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous and nonempty (it contains the origin $0_{\mathbb{R}^{n}}$), and with the penalty $P=V$. By The Hamilton-Jacobi Equation for Controlled Langevin Dynamics in a Potential on Euclidean Space §operator, $F$ is the penalty-drift Hamilton-Jacobi operator with potential $V$ and the given coefficients, and the dissipation hypothesis Well-Posedness of the Langevin Hamilton-Jacobi Equation on Euclidean Space under a Dissipation Inequality §dissipation is hypothesis Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §dissipation for $P=V$.

It remains to verify hypothesis Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §one-sided. Let $R\in\mathbb{R}$ and $D_{R}=\{x\in\mathbb{R}^{n}:V(x)<R\}$. By Basic Properties of the Sublevel Sets of a Penalty §sublevel-sets, $D_{R}$ is bounded. Since $V$ is of class $C^{2}$ on $\mathbb{R}^{n}$ (Penalty on an Open Subset of Euclidean Space §regularity), The Gradient of a C^2 Function on Euclidean Space is One-Sided Lipschitz on Bounded Sets with $B=D_{R}$ gives a nonnegative $c_{R}$ with $-c_{R}\lVert x-y\rVert^{2}\le(DV(x)-DV(y))\cdot(x-y)$ for all $x,y\in D_{R}$.

Hence all hypotheses of Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality hold, and its claims Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §comparison and Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §well-posed are claims 1 and 2.