Proof of 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

Each result cited is universally quantified over the data in its own statement. Real order and arithmetic are used through the results placed in force by The Real Numbers: Standing Notation and Background §background.

Step 1 (The hypotheses of the general corollary). We apply Well-Posedness of the Hamilton-Jacobi Equation with Gibbs Score Drift for a Semiconvex Cylindrical Potential: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution 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 for the admissible split $\varsigma$ fixed in the statement: the space $X=H^{-m}(\mathbb{T}^{n})$, its orthonormal basis $(e_{j})_{j\in\mathbb{N}}$, the noise weights $a$ with bound $1$ and the variance sequence $c^{(\varsigma)}$ in place of $c$, which is a variance sequence 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 §split; so the reference measure of that corollary is $\rho=\gamma_{c^{(\varsigma)}}$, that of the statement. The data of that corollary are instantiated as follows.

(a) $K:=K_{\varsigma}$, the greater of $0$ and $1-\varsigma^{-1}$, which is nonnegative, and $V:=V_{\varsigma}$: 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 §potential, $V_{\varsigma}$ is an admissible cylindrical potential with head dimension $d_{M}$, profile $v_{\varsigma}$ and semiconvexity constant $K_{\varsigma}$. That corollary assumes no relation between $K$ and $\beta/\kappa$.

(b) $\beta:=1$, a positive real number, and the positive constant of that corollary (written $\kappa$ there, a letter used here for the enumeration) $:=\bar{r}_{\varsigma}$, the constant 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 §split; it is at least $1$, being the greatest of $1$ and the numbers $r^{(\varsigma)}_{k}$, hence positive. 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 §split, $c^{(\varsigma)}_{j}\le\bar{r}_{\varsigma}\,a_{j}$ for every $j\in\mathbb{N}$, which is the required bound.

(c) With these data, the Gibbs entropy pair of that corollary is the pair with potential $V_{\varsigma}$ and temperature $1$ relative to $\gamma_{c^{(\varsigma)}}$, whose hypothesis holds with the constant $\bar{r}_{\varsigma}$; 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 it is the quadruple $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ of the statement, with $\mathcal{D}$ the set of the $\mu\in\mathcal{P}(X)$ of finite relative entropy with respect to $\gamma^{\mathrm{HS}}_{J,\eta}$. Its Gibbs measure $\gamma^{V_{\varsigma}}_{1}$ equals $\gamma^{\mathrm{HS}}_{J,\eta}$ 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 §gibbs.

(d) The numbers $\lambda_{0},\theta$, the function $g$ and the constant $C$ of 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 §data satisfy exactly the requirements of Well-Posedness of the Hamilton-Jacobi Equation with Gibbs Score Drift for a Semiconvex Cylindrical Potential: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §data: $0<\lambda_{0}$, $0<\theta\le1$, $g$ bounded and uniformly continuous on $\mathcal{D}$ relative to $\mathcal{D}$ in the same 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 with $\rho=\gamma_{c^{(\varsigma)}}$ (the same for every admissible split, since $\mathcal{P}^{a}_{\rho}$ is 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 and $W_{a}$ depends only on $a$, by The Noise Wasserstein Distance §distance), with $\mathbb{R}$ carrying the metric of The Absolute Value Metric on the Real Line, and $0\le C$ with $|g(\mu)|\le C$ for every $\mu\in\mathcal{D}$.

(e) The viscosity subsolutions, supersolutions and solutions of the statement are those of The Hamilton-Jacobi Equation of the Hubbard-Stratonovich Ising Field on the Lattice Torus with White Noise §equation with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$, formed with the split $\varsigma$; by that clause they are those of the Hamilton-Jacobi equation with Gibbs score drift relative to $\gamma^{V_{\varsigma}}_{1}$, with temperature $1$, the constant $\bar{r}_{\varsigma}$, discount $\lambda_{0}$, control cost $\theta$ and running cost $g$. These are precisely the viscosity notions of Well-Posedness of the Hamilton-Jacobi Equation with Gibbs Score Drift for a Semiconvex Cylindrical Potential: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution for the data (a) to (d).

Step 2 (Comparison; claim 1). Let $u,w:\mathcal{D}\to\mathbb{R}$ and $b,b'\in\mathbb{R}$ be as in claim 1. The hypotheses of Well-Posedness of the Hamilton-Jacobi Equation with Gibbs Score Drift for a Semiconvex Cylindrical Potential: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §comparison are that $u(\mu)\le b$ and $b'\le w(\mu)$ for every $\mu\in\mathcal{D}$ and that $u$ is a viscosity subsolution and $w$ a viscosity supersolution; the penalty-subordinate growth of $u$ and $w$ mentioned there is not an extra hypothesis but a consequence that clause derives from the bounds. All of these hold for the data of Step 1, so that clause gives $u(\mu)\le w(\mu)$ for every $\mu\in\mathcal{D}$.

Step 3 (Existence; claim 2). Well-Posedness of the Hamilton-Jacobi Equation with Gibbs Score Drift for a Semiconvex Cylindrical Potential: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §existence, for the data of Step 1 and the constant $C$, gives a viscosity solution $u:\mathcal{D}\to\mathbb{R}$ with $-\lambda_{0}^{-1}C\le u(\mu)\le\lambda_{0}^{-1}C$ for every $\mu\in\mathcal{D}$.

Step 4 (Uniqueness; claim 3). Let $u,u'$ be bounded viscosity solutions. Well-Posedness of the Hamilton-Jacobi Equation with Gibbs Score Drift for a Semiconvex Cylindrical Potential: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §uniqueness, for the data of Step 1, gives $u(\mu)=u'(\mu)$ for every $\mu\in\mathcal{D}$. The solution of Step 3 is bounded, by the constant $\lambda_{0}^{-1}C$, so the equation has exactly one bounded viscosity solution.