Proof of 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

Each result cited is universally quantified over the data in its own statement. Elementary ordered-field arithmetic and order, and the properties of finite sums, are used without further comment, as provided by The Real Numbers: Standing Notation and Background §background.

The data. By The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §gaussian, the notation of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation is in force with the reference measure $\rho=\gamma_{c}$. 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, $c_{j}\le1\cdot a_{j}$ for every $j\in\mathbb{N}$, so the temperature $\beta=1$ and the constant $\kappa=1$ satisfy the standing hypotheses of Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift and a Score-Paired Wick-Square Cost on a Hilbert Space, by Gaussian Dressing, 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 and The Score-Paired Wick-Square Cost is the Wick-Ordered Series, and Cutoff Counterterms Are Forced up to a Convergent Constant, with the Gaussian entropy pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ of the corollary. Since $n\le3$, $w$ is 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. By hypothesis $0<\lambda_{0}$, $0<\theta\le1$ and $\varepsilon_{0}=1+\lambda_{0}+2\theta\min\{w_{0},0\}>0$, and 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 gives $a_{j}/c_{j}+\lambda_{0}+2\theta w_{j}c_{j}\ge\varepsilon_{0}$ for every $j$; as $\beta=1$, this is the inequality $\beta a_{j}/c_{j}+\lambda_{0}+2\theta w_{j}c_{j}/\beta\ge\varepsilon_{0}$ required in Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift and a Score-Paired Wick-Square Cost on a Hilbert Space, by Gaussian Dressing and 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. The running cost $g$ is bounded and uniformly continuous on $\mathcal{D}$, relative to the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$, with $0\le C$ and $|g|\le C$ on $\mathcal{D}$. So Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift and a Score-Paired Wick-Square Cost on a Hilbert Space, by Gaussian Dressing applies to the data $\beta=1$, $\kappa=1$, $w$, $\lambda_{0}$, $\theta$, $\varepsilon_{0}$, $g$ and $C$; its profile $\Phi_{0}$ and constant $e$ are those 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 §profile and 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 these data, that is, the $\Phi_{0}$ and $e$ of the corollary; and its viscosity subsolutions, supersolutions and solutions relative to the profile $\Phi_{0}$ are those of The Hamilton-Jacobi Equation with Gaussian Score Drift and a Score-Paired Wick-Square Cost on a Hilbert Space §equation relative to $\gamma_{c}$ with temperature $1$, discount $\lambda_{0}$, control cost $\theta$, Wick couplings $w$ and running cost $g$, which are those of the corollary.

The Wick-ordered terms. Let $\nu\in\mathcal{D}_{\Sigma}$ and $j\in\mathbb{N}$. Since $\nu\in\mathcal{P}^{a}_{\rho}$ (The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §pair), the function $x\mapsto x_{j}^{2}$ is $\nu$-integrable, by the preamble of Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions. For $x\in X$, $x_{j}=\langle x,e_{j}\rangle_{H^{-m}}$ (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates) equals $\rho^{m}_{\kappa(j)}\,x(\kappa(j))$ 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 §basis, so $x_{j}^{2}=a_{j}\,x(\kappa(j))^{2}$ and $x(\kappa(j))^{2}=a_{j}^{-1}x_{j}^{2}$, with $a_{j}>0$ 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 §weights. Hence $x\mapsto x(\kappa(j))^{2}$ is $\nu$-integrable by Linearity and Monotonicity of the Lebesgue Integral §integrable. 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, $w_{j}x_{j}^{2}=w_{0}\,x(\kappa(j))^{2}$ for every $x\in X$, so Linearity and Monotonicity of the Lebesgue Integral §integrable gives

$$w_{j}\int_{X}x_{j}^{2}\,\nu(dx)=\int_{X}w_{j}x_{j}^{2}\,\nu(dx)=\int_{X}w_{0}\,x(\kappa(j))^{2}\,\nu(dx)=w_{0}\int_{X}x(\kappa(j))^{2}\,\nu(dx),\tag{1}$$

and the same clause gives $w_{j}c_{j}=w_{0}\,\mu_{\kappa(j)}^{-1}$. In particular the $j$-th term $w_{j}\bigl(\int_{X}x_{j}^{2}\,d\nu-c_{j}\bigr)$ of the series of The Score-Paired Wick-Square Cost is the Wick-Ordered Series, and Cutoff Counterterms Are Forced up to a Convergent Constant §series is $w_{0}\bigl(\int_{X}x(\kappa(j))^{2}\,\nu(dx)-\mu_{\kappa(j)}^{-1}\bigr)$. Summing (1) and the last identity over $j\le N$, for every $N\in\mathbb{N}$,

$$\sum_{j=1}^{N}w_{j}\int_{X}x_{j}^{2}\,\nu(dx)=w_{0}\sum_{j=1}^{N}\int_{X}x(\kappa(j))^{2}\,\nu(dx),\qquad\sum_{j=1}^{N}w_{j}c_{j}=w_{0}\sum_{j=1}^{N}\mu_{\kappa(j)}^{-1}.\tag{2}$$

Clauses 1, 2 and 3. These are Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift and a Score-Paired Wick-Square Cost on a Hilbert Space, by Gaussian Dressing §comparison, Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift and a Score-Paired Wick-Square Cost on a Hilbert Space, by Gaussian Dressing §existence and Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift and a Score-Paired Wick-Square Cost on a Hilbert Space, by Gaussian Dressing §uniqueness for the data above.

Clause 4. Apply The Score-Paired Wick-Square Cost is the Wick-Ordered Series, and Cutoff Counterterms Are Forced up to a Convergent Constant §counterterms with $\beta=1$, $\kappa=1$, the couplings $w$ and the sequence $(C_{N})_{N\in\mathbb{N}}$; its score-paired Wick-square cost is the $G_{w}$ of the corollary. By (2), its functions $\sum_{j=1}^{N}w_{j}\int_{X}x_{j}^{2}\,\nu(dx)-C_{N}$ are the functions $G_{N}(\nu)$ of clause 4, and its numbers $e_{N}=\sum_{j=1}^{N}w_{j}c_{j}-C_{N}$ are $w_{0}\sum_{j=1}^{N}\mu_{\kappa(j)}^{-1}-C_{N}$. By the equivalence of (ii) and (iii) there, $(G_{N}(\nu))_{N\in\mathbb{N}}$ converges for every $\nu\in\mathcal{D}_{\Sigma}$ if and only if $(e_{N})_{N\in\mathbb{N}}$ converges, and in that case $\lim_{N\to\infty}G_{N}(\nu)=G_{w}(\nu)+e_{\infty}$ for every $\nu\in\mathcal{D}_{\Sigma}$, with $e_{\infty}$ the limit of $(e_{N})_{N\in\mathbb{N}}$. This is clause 4.

Clause 5. Suppose $2\le n$ and $0<w_{0}$. For every $j$, $\mu_{\kappa(j)}^{-1}>0$ (as $1\le\mu_{\kappa(j)}$, see the preamble of 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) and $c_{j}>0$ (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 §variances), so $w_{j}=w_{0}\,\mu_{\kappa(j)}^{-1}/c_{j}>0$ 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 §wick; in particular $w_{j}\ge0$. 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 §divergent, the series $\sum_{j=1}^{\infty}w_{j}c_{j}$ does not converge. So The Score-Paired Wick-Square Cost is the Wick-Ordered Series, and Cutoff Counterterms Are Forced up to a Convergent Constant §bare, applied with $\beta=1$, $\kappa=1$ and $w$, gives for every $\nu\in\mathcal{D}_{\Sigma}$ and $M\in\mathbb{R}$ an $N_{0}\in\mathbb{N}$ with $\sum_{j=1}^{N}w_{j}\int_{X}x_{j}^{2}\,\nu(dx)>M$ for every $N\ge N_{0}$; by (2) the left-hand side is $w_{0}\sum_{j=1}^{N}\int_{X}x(\kappa(j))^{2}\,\nu(dx)$.