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

In the setting of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation, let $N\in\mathbb{N}$, let $\varkappa,\mathfrak{m}\in\mathbb{R}$ satisfy $0<\varkappa$, and let $V_{N}:X\to\mathbb{R}$ be the Galerkin Wick-ordered $\phi^{4}$ potential with cutoff $N$, coupling $\varkappa$ and mass $\mathfrak{m}$, so that, with the Galerkin field $\Phi_{N}[x]$ of $x\in X$, the Wick constant $C_{N}$ and the measure $\lambda_{Q}$ on the cell $Q$ of that definition,

$$V_{N}(x)=\int_{Q}\Bigl(\frac{\varkappa}{4}\bigl(\Phi_{N}[x](z)^{4}-6C_{N}\Phi_{N}[x](z)^{2}+3C_{N}^{2}\bigr)+\frac{\mathfrak{m}}{2}\bigl(\Phi_{N}[x](z)^{2}-C_{N}\bigr)\Bigr)\,\lambda_{Q}(dz)\qquad(x\in X).$$

No condition is placed on the enumeration $\kappa$. $V_{N}$ is an admissible cylindrical potential, with head dimension the Galerkin head dimension $d_{N}$, 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. Let $\beta\in\mathbb{R}$ be positive, let $\gamma^{V_{N}}_{\beta}$ be the Gibbs measure of $V_{N}$ at temperature $\beta$, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gibbs entropy pair with potential $V_{N}$ and temperature $\beta$, whose hypothesis holds with the constant $1$, since $c_{j}\le a_{j}$ for every $j\in\mathbb{N}$ 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 §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 $\gamma^{V_{N}}_{\beta}$, by The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §penalty-domain. $W_{a}$ 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, which contains $\mathcal{D}$. $|s|$ 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 with Gibbs score drift relative to $\gamma^{V_{N}}_{\beta}$, whose hypothesis likewise holds with the constant $1$, with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$,

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

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 $\nu\in\mathcal{D}_{\Sigma}$, and $\nabla_{a}V_{N}$ the noise gradient of $V_{N}$, computed in 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 §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.