Well-Posedness of the Hamilton-Jacobi Equation of the Hubbard-Stratonovich Ising Field on the Lattice Torus at Every Cutoff, Interaction and Shift: Comparison, Existence and Uniqueness

In the settings of The Real Numbers: Standing Notation and Background and A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, the latter read as in The Hubbard-Stratonovich Field Law is the Gibbs Measure of an Admissible Cylindrical Potential for Every Admissible Split of Its Gaussian, and the Gibbs Entropy Pair Does Not Depend on the Split with the variance sequence $c^{(\varsigma)}$ of the admissible split $\varsigma$ fixed below, and in the situation of that lemma, with $M\in\mathbb{N}$, the reflection-symmetric periodic pair interaction $\mathfrak{j}$ of side $L=2M+1$, its interaction matrix $J$ and symbol $\hat{J}$, the shift $\eta\in\mathbb{R}$ for $\mathfrak{j}$ and the Hubbard-Stratonovich field law $\gamma^{\mathrm{HS}}_{J,\eta}$ as there; $\mathfrak{q}_{k}=\hat{J}(k)+\eta$ is positive for every $k$ in the cube $\Gamma_{M}$, because $\eta$ is a shift for $\mathfrak{j}$, and no further condition is placed on $M$, $\mathfrak{j}$ or $\eta$. No condition is placed on the enumeration $\kappa$. Let $\varsigma$ be an admissible split, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gibbs entropy pair of The Hubbard-Stratonovich Field Law is the Gibbs Measure of an Admissible Cylindrical Potential for Every Admissible Split of Its Gaussian, and the Gibbs Entropy Pair Does Not Depend on the Split §independence, which is a noise penalty pair on $\mathcal{P}^{a}_{\rho}$ by The Gibbs Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair and does not depend on $\varsigma$; $\mathcal{D}$ is the set of the Borel probability measures on $X=H^{-m}(\mathbb{T}^{n})$ of finite relative entropy with respect to $\gamma^{\mathrm{HS}}_{J,\eta}$. $W_{a}$ 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 with $\rho=\gamma_{c^{(\varsigma)}}$, which contains $\mathcal{D}$ and does not depend on $\varsigma$ by The Hubbard-Stratonovich Field Law is the Gibbs Measure of an Admissible Cylindrical Potential for Every Admissible Split of Its Gaussian, and the Gibbs Entropy Pair Does Not Depend on the Split §independence. $|s|$ is the absolute value of $s\in\mathbb{R}$.

(The data)¶ $\lambda_{0},\theta\in\mathbb{R}$ satisfy $0<\lambda_{0}$ and $0<\theta\le1$, and $g:\mathcal{D}\to\mathbb{R}$ is 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, the larger of $0$ and a bound for $|g|$ serving.

Viscosity subsolutions, supersolutions and solutions are those of the Hamilton-Jacobi equation of the Hubbard-Stratonovich Ising field with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$,

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

an equation in $(\nu,r,q)$ read as in that clause, with $\langle\cdot,\cdot\rangle_{\nu}$ and $\lVert\cdot\rVert_{\nu}$ the inner product and norm of $L^{2}(\nu;X^{a})$, $Z^{a}_{\nu}$ the noise score field of $\nu\in\mathcal{D}_{\Sigma}$ relative to $\gamma_{c^{(\varsigma)}}$, and $\nabla_{a}V_{\varsigma}$ the noise gradient of the potential $V_{\varsigma}$ of The Hubbard-Stratonovich Field Law is the Gibbs Measure of an Admissible Cylindrical Potential for Every Admissible Split of Its Gaussian, and the Gibbs Entropy Pair Does Not Depend on the Split §potential, computed in The Hubbard-Stratonovich Field Law is the Gibbs Measure of an Admissible Cylindrical Potential for Every Admissible Split of Its Gaussian, and the Gibbs Entropy Pair Does Not Depend on the Split §gradient.

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

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

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