The Discounted Hamilton-Jacobi Equation with a Penalty Drift on the Noise Wasserstein Space

In the setting of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a noise penalty pair on $\mathcal{P}^{a}_{\rho}$, let $\lambda_{0}\in\mathbb{R}$ and $\theta\in\mathbb{R}$ be positive, and let $g:\mathcal{D}\to\mathbb{R}$. The bundle $\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$ of noise fields over $\mathcal{D}_{\Sigma}$ is that of The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts §bundle, and $\tfrac{s}{2}$ denotes the product of a real number $s$ with the multiplicative inverse of $2=1+1$, which exists by Elementary Order Arithmetic in an Ordered Field §halving.

1. (The operator)¶ For $(\nu,q)\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$ the score $\Sigma(\nu)$ lies in $T^{a}_{\nu}$, hence in $L^{2}(\nu;X^{a})$, and $\nu\in\mathcal{D}$, by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, so that $\langle\Sigma(\nu),q\rangle_{\nu}$ and $g(\nu)$ are real numbers. The Hamilton-Jacobi operator with penalty drift on the noise Wasserstein space, with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$, is the function

$$F:\ \mathcal{V}^{a}(\mathcal{D}_{\Sigma})\times\mathbb{R}\to\mathbb{R},\qquad F(\nu,r,q)=\lambda_{0}\,r+\frac{\theta}{2}\,\lVert q\rVert_{\nu}^{2}+\langle\Sigma(\nu),q\rangle_{\nu}-g(\nu),$$

a first-order equation operator over $\mathcal{D}_{\Sigma}$.

2. (The equation)¶ The Hamilton-Jacobi equation with penalty drift on the noise Wasserstein space is

$$\lambda_{0}\,r+\frac{\theta}{2}\,\lVert q\rVert_{\nu}^{2}+\langle\Sigma(\nu),q\rangle_{\nu}=g(\nu),$$

an equation in $(\nu,r,q)$ with $(\nu,q)\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$ and $r\in\mathbb{R}$; equivalently, $F(\nu,r,q)=0$ with $F$ the operator of clause 1. A viscosity solution of the equation is a function $u:\mathcal{D}\to\mathbb{R}$ that is a viscosity solution of $F$ relative to the noise penalty pair; viscosity sub- and supersolutions of the equation are read in the same way.