Proof of Well-Posedness of the Hamilton-Jacobi Equation with the Free-Field Score Drift for Laws of Distributions on the Torus: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution

Each result cited is universally quantified over the data in its own statement.

By The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §gaussian the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation is in force with the space $X=H^{-m}(\mathbb{T}^{n})$, its orthonormal basis $(e_{j})_{j\in\mathbb{N}}$, the noise weights $a$ and the variance sequence $c$ of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §data, and with reference measure $\rho=\gamma_{c}$, the free field. We apply Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution in this setting with temperature $\beta=1$ and with its constant equal to $1$: both are positive, and $c_{k}\le1\cdot a_{k}$ for every $k\in\mathbb{N}$ 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, so the hypothesis of that corollary holds, and the Gaussian entropy pair with temperature $1$ it names is the pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ of the present statement.

The remaining data of that corollary are taken to be those of the present statement, and they satisfy its hypotheses verbatim: $\lambda_{0},\theta\in\mathbb{R}$ with $0<\lambda_{0}$ and $0<\theta\le1$; $g:\mathcal{D}\to\mathbb{R}$ 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, the noise Wasserstein distance $W_{a}$ and the set $\mathcal{P}^{a}_{\rho}$ being those of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §background for $\rho=\gamma_{c}$ in both statements, and with the same metric on $\mathbb{R}$ (The Absolute Value Metric on the Real Line); and the same $C\in\mathbb{R}$ with $0\le C$ and $|g(\mu)|\le C$ for every $\mu\in\mathcal{D}$.

The viscosity notions coincide. In both statements a viscosity subsolution, supersolution or solution is one of the Hamilton-Jacobi equation with Gaussian score drift relative to $\gamma_{c}$ with temperature $1$, discount $\lambda_{0}$, control cost $\theta$ and running cost $g$, that is, by The Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space §equation, a viscosity subsolution, supersolution or solution of the one operator $F$ of The Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space §operator with these data, relative to the one Gaussian entropy pair. The equation displayed in the present statement is the equation of The Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space §equation with $\beta=1$, since $1\cdot\langle Z^{a}_{\nu},q\rangle_{\nu}=\langle Z^{a}_{\nu},q\rangle_{\nu}$.

Since the data, the hypotheses and the notion of solution agree, claims 1, 2 and 3 are exactly Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §comparison, Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §existence and Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §uniqueness for these data, with the same bounds $b,b'$ in claim 1, the same bounds $\pm\lambda_{0}^{-1}C$ in claim 2, and the same conclusion of exactly one bounded viscosity solution in claim 3.