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 $\kappa\in\mathbb{R}$ be nonnegative, 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 §l2mu, 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 and $\mathrm{tr}\,Y$ the trace of $Y\in\mathcal{S}(d)$; 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 here, the dimension written $q$ in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces and in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §background not being used. Of the penalty pair only $\mathcal{D}_{\Sigma}$ and $\Sigma$ enter the operator; $\mathcal{D}$ and $\mathcal{E}$ enter its $\delta$-shifts and the envelopes through which the equation is read.

1. (The operator)¶ 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 §pair. The Hamilton-Jacobi operator with common noise and penalty drift, with discount $\lambda_{0}$, common-noise intensity $\kappa$, control cost $\theta$ and running cost $g$, is the function

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

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

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. For a function $u:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ it is read as follows. If $(\Omega,\mathcal{F},P)$ is rich and $u$ is a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$, with intrinsic gradient $\nabla u(\nu)$ and translation Hessian $H_{u}(\nu)$, the equation holds classically at $\nu\in\mathcal{D}_{\Sigma}$ if it holds with $r=u(\nu)$, $q=\nabla u(\nu)$ and $Y=H_{u}(\nu)$. For a general $u$ and a rich $(\Omega,\mathcal{F},P)$, the equation is understood in the viscosity sense relative to the penalty pair, a solution being a viscosity solution of $F$ relative to the penalty pair, under the remaining hypotheses of that definition.