The Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $c$ be a variance vector, let $a\in\mathbb{R}$ be positive, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian free-energy pair with variances $c$ and temperature $a$, a penalty pair by The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §pair; by its definition, $\Sigma(\nu)=a\,\zeta^{c}_{\nu}$ for $\nu\in\mathcal{D}_{\Sigma}$, where $\zeta^{c}_{\nu}\in T_{\nu}$ is the relative score. Let $\lambda_{0},\theta\in\mathbb{R}$ be positive, let $g:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$, and let $0_{1\times d}\in\mathcal{M}_{1\times d}(\mathbb{R})$ be the zero real $1\times d$ matrix. The bundle $\mathcal{V}(\mathcal{D}_{\Sigma})$ and the inner products and norms of the spaces $L^{2}(\nu;\mathbb{R}^{d})$ are those of the setting. In this item the letter $q$ denotes a vector field and the letter $r$ a real number.

1. (The operator)¶ The lifted Ornstein-Uhlenbeck Hamilton-Jacobi operator with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$ is the Hamilton-Jacobi operator with common noise and penalty drift $F$ of this pair with discount $\lambda_{0}$, common-noise matrix $0_{1\times d}$ (so that $p=1$), control cost $\theta$ and running cost $g$. For $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$ and $Y\in\mathcal{S}(d)$ the matrix $0_{1\times d}^{\top}0_{1\times d}Y$ is the zero matrix, whose trace is $0$, and $\langle a\,\zeta^{c}_{\nu},q\rangle_{\nu}=a\,\langle\zeta^{c}_{\nu},q\rangle_{\nu}$ by bilinearity of the inner product; so

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

2. (The equation)¶ The lifted Ornstein-Uhlenbeck Hamilton-Jacobi equation is

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

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 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 free-energy pair.