The Hamilton-Jacobi Equation of the Hubbard-Stratonovich Ising Field on the Lattice Torus with White Noise

In the situation 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, with $X=H^{-m}(\mathbb{T}^{n})$ and the noise weights $a_{j}$, let $M\in\mathbb{N}$, let $\mathfrak{j}$ be a reflection-symmetric periodic pair interaction of side $L=2M+1$ with interaction matrix $J$, let $\eta$ be a shift for $\mathfrak{j}$, let $\gamma^{\mathrm{HS}}_{J,\eta}$ be the Hubbard-Stratonovich field law, and let $\varsigma$ be an admissible split, with $c^{(\varsigma)}$ and $\bar{r}_{\varsigma}$ 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 §split, $V_{\varsigma}$ 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 §potential, and the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation read as in that lemma, so that $\rho=\gamma_{c^{(\varsigma)}}$. 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 and 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, $V_{\varsigma}$ is an admissible cylindrical potential whose Gibbs measure $\gamma^{V_{\varsigma}}_{1}$ at temperature $1$ is $\gamma^{\mathrm{HS}}_{J,\eta}$. Let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gibbs entropy pair with potential $V_{\varsigma}$ and temperature $1$, whose hypothesis holds with the constant $\bar{r}_{\varsigma}$; it 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 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 same quadruple for every admissible split, $\mathcal{D}$ being the set of the Borel probability measures on $X$ of finite relative entropy with respect to $\gamma^{\mathrm{HS}}_{J,\eta}$. For $\nu\in\mathcal{D}_{\Sigma}$, the noise score field $Z^{a}_{\nu}$ of $\nu$ relative to $\gamma_{c^{(\varsigma)}}$ exists, $Z^{a}_{\nu}$ and the class of $\nabla_{a}V_{\varsigma}$ lie in the noise tangent space $T^{a}_{\nu}\subseteq L^{2}(\nu;X^{a})$ by The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §score-domain, and $\Sigma(\nu)=Z^{a}_{\nu}+\nabla_{a}V_{\varsigma}$ by The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §pair; here $\nabla_{a}V_{\varsigma}(x)=\mathcal{E}_{M}(f_{\varsigma}'\circ\varphi_{x})$ is the noise gradient 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 §gradient, $\mathcal{E}_{M}$ being the white-noise embedding and $\varphi_{x}$ the site field of $x\in X$. Let $\lambda_{0},\theta\in\mathbb{R}$ be positive and let $g:\mathcal{D}\to\mathbb{R}$. $\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$ is the bundle of noise fields over $\mathcal{D}_{\Sigma}$; $\langle\cdot,\cdot\rangle_{\nu}$ and $\lVert\cdot\rVert_{\nu}$ are the inner product and norm of $L^{2}(\nu;X^{a})$, as in The Hamilton-Jacobi Equation with Gibbs Score Drift on a Hilbert Space; and $\frac{\theta}{2}=\theta\cdot2^{-1}$, where $2^{-1}$ exists by Elementary Order Arithmetic in an Ordered Field §halving. In this item the letter $q$ denotes a noise field and the letter $r$ a real number.

1. (The operator)¶ The Hamilton-Jacobi operator of the Hubbard-Stratonovich Ising field with discount $\lambda_{0}$, control cost $\theta$ and running cost $g$ is the Hamilton-Jacobi operator with Gibbs score drift $F$ relative to $\gamma^{V_{\varsigma}}_{1}=\gamma^{\mathrm{HS}}_{J,\eta}$, with temperature $1$, the constant $\bar{r}_{\varsigma}$, discount $\lambda_{0}$, control cost $\theta$ and running cost $g$:

$$F(\nu,r,q)=\lambda_{0}\,r+\frac{\theta}{2}\,\lVert q\rVert_{\nu}^{2}+\langle\Sigma(\nu),q\rangle_{\nu}-g(\nu)\qquad\bigl((\nu,q)\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma}),\ r\in\mathbb{R}\bigr).$$

$F$ is the operator of the 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 the same quadruple for every admissible split.

2. (The equation)¶ The Hamilton-Jacobi equation of the Hubbard-Stratonovich Ising field is

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

an equation in $(\nu,r,q)$ with $(\nu,q)\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$ and $r\in\mathbb{R}$; equivalently $F(\nu,r,q)=0$, since $\langle\Sigma(\nu),q\rangle_{\nu}=\langle Z^{a}_{\nu},q\rangle_{\nu}+\langle\nabla_{a}V_{\varsigma},q\rangle_{\nu}$ by bilinearity of the inner product. Its viscosity subsolutions, supersolutions and solutions are those of the Hamilton-Jacobi equation with Gibbs score drift relative to $\gamma^{V_{\varsigma}}_{1}$ with these data.