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

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\lambda_{0}\in\mathbb{R}$ and $\theta\in\mathbb{R}$ be positive, let $p\in\mathbb{N}$, let $\Gamma\in\mathcal{M}_{p\times d}(\mathbb{R})$ be a real $p\times d$ matrix, with transpose $\Gamma^{\top}$, and let $g:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$. The bundle $\mathcal{V}(\mathcal{D}_{\Sigma})$ of vector fields over $\mathcal{D}_{\Sigma}$ is that of that clause; for $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the inner product $\langle\cdot,\cdot\rangle_{\nu}$ and norm $\lVert\cdot\rVert_{\nu}$ of $L^{2}(\nu;\mathbb{R}^{d})$ are those of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii), and $T_{\nu}\subseteq L^{2}(\nu;\mathbb{R}^{d})$ is the tangent space; $\mathcal{S}(d)$ is the set of symmetric real $d\times d$ matrices for $Y\in\mathcal{S}(d)$, $\mathrm{tr}(\Gamma^{\top}\Gamma Y)$ is the trace of the matrix product $\Gamma^{\top}\Gamma Y\in\mathcal{M}_{d}(\mathbb{R})$, which is unambiguous by Associativity of the Matrix Product; and $\tfrac{s}{2}$ denotes the product of a real number $s$ with the multiplicative inverse of $2=1+1$, which exists by claim 8 of Elementary Order Arithmetic in an Ordered Field. The letter $q$ denotes a vector field and the letter $p$ the number of rows of $\Gamma$ here; the dimensions written $p$ and $q$ in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (i) and in Euclidean Space and Real Matrices: Standing Notation (ii), and the dimension written $q$ in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (i), are not used.

1. (The operator)(i) For $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$ the score $\Sigma(\nu)$ lies in $T_{\nu}$, hence in $L^{2}(\nu;\mathbb{R}^{d})$, so that $\langle\Sigma(\nu),q\rangle_{\nu}$ is a real number, and $g(\nu)$ is a real number because $\nu\in\mathcal{D}_{\Sigma}\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (i). The Hamilton-Jacobi operator with common noise and penalty drift, with discount $\lambda_{0}$, common-noise matrix $\Gamma$, control cost $\theta$ and running cost $g$, is the function

$$F:\ \mathcal{V}(\mathcal{D}_{\Sigma})\times\mathbb{R}\times\mathcal{S}(d)\to\mathbb{R},$$ $$F(\nu,r,q,Y)=\lambda_{0}\,r-\frac{1}{2}\,\mathrm{tr}\bigl(\Gamma^{\top}\Gamma Y\bigr)+\frac{\theta}{2}\,\lVert q\rVert_{\nu}^{2}+\langle\Sigma(\nu),q\rangle_{\nu}-g(\nu),$$

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

2. (The equation)(ii) The Hamilton-Jacobi equation with common noise and penalty drift is

$$\lambda_{0}\,r-\frac{1}{2}\,\mathrm{tr}\bigl(\Gamma^{\top}\Gamma Y\bigr)+\frac{\theta}{2}\,\lVert q\rVert_{\nu}^{2}+\langle\Sigma(\nu),q\rangle_{\nu}=g(\nu),$$

an equation in $(\nu,r,q,Y)$ with $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$ and $Y\in\mathcal{S}(d)$; equivalently, $F(\nu,r,q,Y)=0$ with $F$ the operator of clause 1. A classical solution of the equation is a classical solution of $F$ on $\mathcal{D}_{\Sigma}$, and 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 penalty pair; classical sub- and supersolutions and viscosity sub- and supersolutions of the equation are read in the same way.