Well-Posedness of the Hamilton-Jacobi Equation with the Free-Field Score Drift for Laws of Distributions on the Torus: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution

In the setting of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian entropy pair with temperature $1$, whose hypothesis holds with the constant $1$ by White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §ratio; thus $\mathcal{D}$ is the set of Borel probability measures on $X=H^{-m}(\mathbb{T}^{n})$ of finite relative entropy with respect to the free field $\gamma_{c}$, by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain. $W_{a}$ is the noise Wasserstein distance, whose cost $|y-x|_{a}^{2}$ equals $\lVert U\rVert_{L^{2}}^{2}$ when $y-x=\hat{U}$ with $U\in L^{2}(\mathbb{T}^{n})$, by White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §noise-space, and $(\mathcal{P}^{a}_{\rho},W_{a})$ is the metric space of The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §metric, which contains $\mathcal{D}$. Let $|s|$ denote the absolute value of $s\in\mathbb{R}$. Let $\lambda_{0},\theta\in\mathbb{R}$ satisfy $0<\lambda_{0}$ and $0<\theta\le1$, and let $g:\mathcal{D}\to\mathbb{R}$ be bounded and uniformly continuous on $\mathcal{D}$, relative to $\mathcal{D}$ in $(\mathcal{P}^{a}_{\rho},W_{a})$ and with $\mathbb{R}$ carrying the metric of The Absolute Value Metric on the Real Line. Fix $C\in\mathbb{R}$ with $0\le C$ and $|g(\mu)|\le C$ for every $\mu\in\mathcal{D}$; such a $C$ exists, since if $C'$ bounds $|g|$ on $\mathcal{D}$ as in Bounded Real-Valued Function on a Set, the larger of $C'$ and $0$ serves. Viscosity subsolutions, supersolutions and solutions are those of the Hamilton-Jacobi equation with Gaussian score drift relative to the free field with temperature $1$ (so that its score term is $1\cdot\langle Z^{a}_{\nu},q\rangle_{\nu}=\langle Z^{a}_{\nu},q\rangle_{\nu}$), discount $\lambda_{0}$, control cost $\theta$ and running cost $g$,

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

an equation in $(\nu,r,q)$ read as in that clause, with $Z^{a}_{\nu}$ the noise score field of $\nu\in\mathcal{D}_{\Sigma}$.

1. (Comparison)¶ Let $u,v:\mathcal{D}\to\mathbb{R}$ and $b,b'\in\mathbb{R}$ satisfy $u(\mu)\le b$ and $b'\le v(\mu)$ for every $\mu\in\mathcal{D}$, let $u$ be a viscosity subsolution and $v$ a viscosity supersolution. Then $u(\mu)\le v(\mu)$ for every $\mu\in\mathcal{D}$.

2. (Existence)¶ There is a viscosity solution $u:\mathcal{D}\to\mathbb{R}$ with, $\lambda_{0}^{-1}$ denoting the multiplicative inverse of $\lambda_{0}$,

$$-\lambda_{0}^{-1}C\le u(\mu)\le\lambda_{0}^{-1}C\qquad\text{for every }\mu\in\mathcal{D}.$$

3. (Uniqueness)¶ Let $u,u':\mathcal{D}\to\mathbb{R}$ be viscosity solutions, each bounded. Then $u(\mu)=u'(\mu)$ for every $\mu\in\mathcal{D}$. Consequently, together with claim 2, the equation has exactly one bounded viscosity solution $u:\mathcal{D}\to\mathbb{R}$.