Well-Posedness of the Hamilton-Jacobi Equation with the Free-Field Score Drift and the Wick-Square Cost for Laws of Distributions on the Torus of Dimension at Most Three, and the Forced Counterterm

In the setting of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation, suppose $n\le3$, 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. Let $w_{0}\in\mathbb{R}$, let $w=(w_{j})_{j\in\mathbb{N}}$ with $w_{j}=w_{0}\,\mu_{\kappa(j)}^{m}$, a sequence of Wick couplings with bound $|w_{0}|$ by The Wick Square of the Free Field on the Torus as Diagonal Couplings: Summability in Dimension at Most Three, the Riccati Condition, and the Divergent Wick Constant §summability, and let $\Pi_{w}$ and $G_{w}$ be its Wick-square corrector and score-paired Wick-square cost; by The Wick Square of the Free Field on the Torus as Diagonal Couplings: Summability in Dimension at Most Three, the Riccati Condition, and the Divergent Wick Constant §wick, the $j$-th term $w_{j}(\int x_{j}^{2}\,d\nu-c_{j})$ of its series in The Score-Paired Wick-Square Cost is the Wick-Ordered Series, and Cutoff Counterterms Are Forced up to a Convergent Constant §series is $w_{0}\bigl(\int x(\kappa(j))^{2}\,\nu(dx)-\mu_{\kappa(j)}^{-1}\bigr)$. Let $|s|$ denote the absolute value of $s\in\mathbb{R}$ and $\min\{s,s'\}$ the minimum. Let $\lambda_{0},\theta\in\mathbb{R}$ satisfy $0<\lambda_{0}$, $0<\theta\le1$ and $1+\lambda_{0}+2\theta\min\{w_{0},0\}>0$, and let $g:\mathcal{D}\to\mathbb{R}$ be bounded and uniformly continuous on $\mathcal{D}$, relative to $\mathcal{D}$ in the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ 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 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 $\mu\in\mathcal{D}$. Let $\Phi_{0}$ be the quadratic profile and $e$ the real number of Gaussian Dressing of the Score-Paired Wick-Square Cost on a Hilbert Space: Riccati Coefficients, Dressed Variances, the Quadratic Profile, the Constant, the Dressed Pair and the Shifted Operator §constant, for temperature $1$, these $w$, $\lambda_{0}$, $\theta$, and $\varepsilon_{0}=1+\lambda_{0}+2\theta\min\{w_{0},0\}$, which satisfies its hypothesis by The Wick Square of the Free Field on the Torus as Diagonal Couplings: Summability in Dimension at Most Three, the Riccati Condition, and the Divergent Wick Constant §riccati. Viscosity subsolutions, supersolutions and solutions relative to the profile $\Phi_{0}$ are those of the Hamilton-Jacobi equation with Gaussian score drift and the Wick-square cost relative to the free field, with temperature $1$, discount $\lambda_{0}$, control cost $\theta$, Wick couplings $w$ and running cost $g$, which reads

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

for a noise field $q\in L^{2}(\nu;X^{a})$ over $\nu\in\mathcal{D}_{\Sigma}$ and a real number $r$, with $\langle\cdot,\cdot\rangle_{\nu}$ and $\lVert\cdot\rVert_{\nu}$ the inner product and norm of $L^{2}(\nu;X^{a})$ and $Z^{a}_{\nu}$ the noise score field, as in that clause. For $x\in X$ and $k\in\mathbb{Z}^{n}$, $x(k)$ is the value at $k$ of the coefficient family $x$, as in The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §data. For $u:\mathcal{D}\to\mathbb{R}$, $u-\Phi_{0}$ is the function on $\mathcal{D}$ with value $u(\mu)-\Phi_{0}(\mu)$ at $\mu$, and $\lambda_{0}^{-1}$ is the multiplicative inverse of $\lambda_{0}$.

1. (Comparison)¶ Let $u:\mathcal{D}\to\mathbb{R}$ be a viscosity subsolution and $v:\mathcal{D}\to\mathbb{R}$ a viscosity supersolution relative to the profile $\Phi_{0}$, and let $B_{1},B_{2}\in\mathbb{R}$ satisfy $u(\mu)-\Phi_{0}(\mu)\le B_{1}$ and $B_{2}\le v(\mu)-\Phi_{0}(\mu)$ for every $\mu\in\mathcal{D}$. Then $u(\mu)\le v(\mu)$ for every $\mu\in\mathcal{D}$.

2. (Existence)¶ There is a viscosity solution $U:\mathcal{D}\to\mathbb{R}$ relative to the profile $\Phi_{0}$ with $-\lambda_{0}^{-1}(C+|e|)\le U(\mu)-\Phi_{0}(\mu)\le\lambda_{0}^{-1}(C+|e|)$ for every $\mu\in\mathcal{D}$.

3. (Uniqueness)¶ Let $U,U':\mathcal{D}\to\mathbb{R}$ be viscosity solutions relative to the profile $\Phi_{0}$ such that $U-\Phi_{0}$ and $U'-\Phi_{0}$ are bounded. Then $U(\mu)=U'(\mu)$ for every $\mu\in\mathcal{D}$.

4. (The counterterm is forced)¶ Let $(C_{N})_{N\in\mathbb{N}}$ be a real sequence, and for $\nu\in\mathcal{D}_{\Sigma}$ and $N\in\mathbb{N}$ let $G_{N}(\nu)=w_{0}\sum_{j=1}^{N}\int_{X}x(\kappa(j))^{2}\,\nu(dx)-C_{N}$. Then the sequence $(G_{N}(\nu))_{N\in\mathbb{N}}$ converges for every $\nu\in\mathcal{D}_{\Sigma}$ if and only if the sequence $\bigl(w_{0}\sum_{j=1}^{N}\mu_{\kappa(j)}^{-1}-C_{N}\bigr)_{N\in\mathbb{N}}$ converges, and then $\lim_{N\to\infty}G_{N}(\nu)=G_{w}(\nu)+e_{\infty}$ for every $\nu\in\mathcal{D}_{\Sigma}$, where $e_{\infty}$ is the limit of the latter sequence.

5. (The bare cost diverges)¶ Suppose $2\le n$ and $0<w_{0}$. Then for every $\nu\in\mathcal{D}_{\Sigma}$ and every $M\in\mathbb{R}$ there is $N_{0}\in\mathbb{N}$ with $w_{0}\sum_{j=1}^{N}\int_{X}x(\kappa(j))^{2}\,\nu(dx)>M$ for every $N\in\mathbb{N}$ with $N\ge N_{0}$.