The Lifted Hamilton-Jacobi Equation with a Wick-Square Cost 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, so that $\Sigma(\nu)=a\,\zeta^{c}_{\nu}$ for $\nu\in\mathcal{D}_{\Sigma}$, where $\zeta^{c}_{\nu}$ is the relative score. Let $\lambda_{0},\theta\in\mathbb{R}$ be positive, let $\beta\in\mathbb{R}^{d}$, let $g:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$, and let $V_{c,\beta}$ be the Wick-square potential with variances $c$ and couplings $\beta$. By Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic, applied to the diagonal matrix with diagonal entries $2\beta_{1},\dots,2\beta_{d}$, linear coefficient $0$ and constant term $-\sum_{i=1}^{d}\beta_{i}c_{i}$, the function $V_{c,\beta}$ is of class $C^{2}$ on $\mathbb{R}^{d}$ with $\partial_{j}\partial_{i}V_{c,\beta}(x)$ equal to $2\beta_{i}$ if $i=j$ and to $0$ otherwise; so Integrals of Functions with a Bounded Hessian are Intrinsic Test Functions on the Wasserstein Space §integrable, with $M=2\sum_{i=1}^{d}|\beta_{i}|$, makes $V_{c,\beta}$ integrable with respect to every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, and

$$G:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R},\qquad G(\nu)=\int_{\mathbb{R}^{d}}V_{c,\beta}\,d\nu+g(\nu),$$

is defined. 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 Wick-square Hamilton-Jacobi operator with discount $\lambda_{0}$, control cost $\theta$, couplings $\beta$ and running cost $g$ is the lifted Ornstein-Uhlenbeck Hamilton-Jacobi operator $F$ of the pair with discount $\lambda_{0}$, control cost $\theta$ and running cost $G$; thus

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

for $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$ and $Y\in\mathcal{S}(d)$.

2. (The equation)¶ The lifted Wick-square Hamilton-Jacobi equation is

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

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. For an intrinsic test function $\Phi$ on $\mathcal{D}$, a viscosity subsolution, supersolution or solution of the equation relative to the profile $\Phi$ 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 and the profile $\Phi$.