The Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, so that the reference measure is $\rho=\gamma_{c}$, let $\beta$ and $\kappa$ be positive real numbers with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian entropy pair with temperature $\beta$, whose hypothesis holds with this $\kappa$; it is a noise penalty pair on $\mathcal{P}^{a}_{\rho}$ by The Gaussian Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair, and $\Sigma(\nu)=\beta\,Z^{a}_{\nu}$ for $\nu\in\mathcal{D}_{\Sigma}$, where $Z^{a}_{\nu}$ is the noise score field. Let $\lambda_{0},\theta\in\mathbb{R}$ 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 that definition, $\tfrac{s}{2}$ 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, and $\langle\cdot,\cdot\rangle_{\nu}$ and $\lVert\cdot\rVert_{\nu}$ are the inner product and norm of $L^{2}(\nu;X^{a})$. In this item the letter $q$ denotes a noise field and the letter $r$ a real number.

1. (The operator)¶ The Hamilton-Jacobi operator with Gaussian score drift relative to $\gamma_{c}$, with temperature $\beta$, discount $\lambda_{0}$, control cost $\theta$ and running cost $g$, is the Hamilton-Jacobi operator with penalty drift $F$ of this pair with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$. For $(\nu,q)\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$ and $r\in\mathbb{R}$ one has $\langle\beta\,Z^{a}_{\nu},q\rangle_{\nu}=\beta\,\langle Z^{a}_{\nu},q\rangle_{\nu}$ by bilinearity of the inner product, so

$$F(\nu,r,q)=\lambda_{0}\,r+\frac{\theta}{2}\,\lVert q\rVert_{\nu}^{2}+\beta\,\langle Z^{a}_{\nu},q\rangle_{\nu}-g(\nu).$$

2. (The equation)¶ The Hamilton-Jacobi equation with Gaussian score drift relative to $\gamma_{c}$ is

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

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 subsolution, supersolution or solution of the equation is a function $u:\mathcal{D}\to\mathbb{R}$ that is a viscosity subsolution, supersolution or solution of $F$ relative to the Gaussian entropy pair.