Proof of Well-Posedness of the Galerkin $Phi^4$ Stochastic-Quantisation Hamilton-Jacobi Equation on the Torus for Every Cutoff, Coupling and Mass: 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). 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 $X=H^{-m}(\mathbb{T}^{n})$, its orthonormal basis $(e_{j})_{j\in\mathbb{N}}$, the noise weights $a$, the variance sequence $c$ and the reference measure $\rho=\gamma_{c}$; and the notation of The Real Numbers: Standing Notation and Background is in force by A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §background. So the settings 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 are in force with these data. Let $K_{N}=\max\{3\varkappa C_{N}-\mathfrak{m},0\}$ be the number of The Galerkin Wick-Ordered Phi^4 Potential on the Torus is an Admissible Cylindrical Potential, with Semiconvexity Constant the Positive Part of Three Times the Coupling Times the Wick Constant Minus the Mass. By The Galerkin Wick-Ordered Phi^4 Potential on the Torus is an Admissible Cylindrical Potential, with Semiconvexity Constant the Positive Part of Three Times the Coupling Times the Wick Constant Minus the Mass §admissible, $K_{N}$ is nonnegative and $V_{N}$ is an admissible cylindrical potential with head dimension $d_{N}$, profile $v_{N}$ and semiconvexity constant $K_{N}$; we use that corollary with $K=K_{N}$ and $V=V_{N}$. The temperature $\beta$ is positive by hypothesis, and we take the positive constant of that corollary (written $\kappa$ there, a letter reserved for the enumeration here by The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §data) to be $1$: for every $j\in\mathbb{N}$, $c_{j}\le a_{j}=1\cdot a_{j}$ 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 §ratio. With these choices the Gibbs entropy pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ of that corollary is the pair with potential $V_{N}$ and temperature $\beta$ fixed in the statement, whose hypothesis holds with the constant $1$; its Gibbs measure is $\gamma^{V_{N}}_{\beta}$. The numbers $\lambda_{0}$, $\theta$, the function $g$ and the constant $C$ of Well-Posedness of the Galerkin Phi^4 Stochastic-Quantisation Hamilton-Jacobi Equation on the Torus for Every Cutoff, Coupling and Mass: 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, with 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. Finally the viscosity subsolutions, supersolutions and solutions of the statement are those of the Hamilton-Jacobi equation with Gibbs score drift relative to $\gamma^{V_{N}}_{\beta}$, with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$, formed with the constant $1$; the displayed equation of the statement is the equation of The Hamilton-Jacobi Equation with Gibbs Score Drift on a Hilbert Space §equation with $V=V_{N}$, its noise gradient $\nabla_{a}V_{N}$ being that of Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient. These are 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.

Step 2 (Comparison; claim 1). Let $u,w$ and $b,b'$ be as in claim 1. They satisfy 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 for the data of Step 1, which 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.