The Discounted Hamilton-Jacobi Equation with Common Noise and Score 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 $\lambda_{0}\in\mathbb{R}$, $\theta\in\mathbb{R}$ and $\sigma\in\mathbb{R}$ be positive, let $\kappa\in\mathbb{R}$ be nonnegative, and let $g:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$. The set $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ of measures of finite Fisher information and the score $\xi_{\nu}\in T_{\nu}$ of such a measure are those of that definition. The bundle $\mathcal{V}(\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d}))$ of vector fields over $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ 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 and $\mathrm{tr}\,Y$ the trace of $Y\in\mathcal{S}(d)$; $\tfrac{s}{2}$ 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; and $\sigma^{2}=\sigma\sigma$. In this statement $\sigma$ denotes the noise intensity; the swap map written $\sigma$ in Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (ii) is not used here, and the letter $q$ denotes a vector field, the dimension written $q$ in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (i) and in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (i) not being used.

1. (The operator)(i) For $(\nu,q)\in\mathcal{V}(\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d}))$ the score $\xi_{\nu}$ lies in $T_{\nu}$, hence in $L^{2}(\nu;\mathbb{R}^{d})$, so that $\langle\xi_{\nu},q\rangle_{\nu}$ is a real number, and $g(\nu)$ is a real number because $\nu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ by Finite Fisher Information, the Score and the Fisher Information of a Probability Measure (i). The Hamilton-Jacobi operator with common noise and score drift, with discount $\lambda_{0}$, common-noise intensity $\kappa$, control cost $\theta$, noise intensity $\sigma$ and running cost $g$, is the function

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

a second-order equation operator over $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$.

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

$$\lambda_{0}\,r-\frac{\kappa}{2}\,\mathrm{tr}\,Y+\frac{\theta}{2}\,\lVert q\rVert_{\nu}^{2}+\frac{\sigma^{2}}{2}\,\langle\xi_{\nu},q\rangle_{\nu}=g(\nu),$$

an equation in $(\nu,r,q,Y)$ with $(\nu,q)\in\mathcal{V}(\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d}))$, $r\in\mathbb{R}$ 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 $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$. In the viscosity sense the equation is read relative to a penalty pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ with $\mathcal{D}_{\Sigma}=\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ and $\Sigma(\nu)=\tfrac{\sigma^{2}}{2}\,\xi_{\nu}$, for which $F$ is the Hamilton-Jacobi operator with common noise and penalty drift of that pair with discount $\lambda_{0}$, common-noise intensity $\kappa$, control cost $\theta$ and running cost $g$: a viscosity solution of the equation is then a function $u:\mathcal{D}\to\mathbb{R}$ that is a viscosity solution of $F$ relative to that pair; classical sub- and supersolutions and viscosity sub- and supersolutions of the equation are read in the same way.