The Hamilton-Jacobi Equation with Gibbs 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 $V$ be an admissible cylindrical potential with noise gradient $\nabla_{a}V$, 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 Gibbs entropy pair with potential $V$ and temperature $\beta$, whose hypothesis holds with this $\kappa$; it is a noise penalty pair on $\mathcal{P}^{a}_{\rho}$ by The Gibbs Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair, and $\Sigma(\nu)=\beta\,Z^{a}_{\nu}+\nabla_{a}V$ for $\nu\in\mathcal{D}_{\Sigma}$, where $Z^{a}_{\nu}$ is the noise score field of $\nu$, which has the required relative score and Fisher information by The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §score-domain. 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 Gibbs score drift relative to $\gamma^{V}_{\beta}$, with 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 $(\nu,q)\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$ the field $Z^{a}_{\nu}$ and the class of $\nabla_{a}V$ lie in the noise tangent space $T^{a}_{\nu}\subseteq L^{2}(\nu;X^{a})$ by The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §score-domain, so for $r\in\mathbb{R}$ one has $\langle\beta Z^{a}_{\nu}+\nabla_{a}V,q\rangle_{\nu}=\beta\langle Z^{a}_{\nu},q\rangle_{\nu}+\langle\nabla_{a}V,q\rangle_{\nu}$ by bilinearity of the inner product, so

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

2. (The equation)¶ The Hamilton-Jacobi equation with Gibbs score drift relative to $\gamma^{V}_{\beta}$ is

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

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 Gibbs entropy pair.