Proof of Well-Posedness of the Lifted N-Particle Hamilton-Jacobi Equation: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution

Each result cited is universally quantified over the data in its own statement, and is applied to the data named at the point of use.

Step 1 (the data at the configuration level). By The N-Particle Potential of a Confining Potential is a Confining Potential on the Configuration Space §confining, the $N$-particle potential $V_{N}$ of $V$ is a confining potential on $\mathbb{R}^{dN}$. By The N-Particle Hamilton-Jacobi Equation with Individual and Common Noise on the Configuration Space §common-noise, the $N$-particle common-noise matrix $\Gamma_{N}$ of $\Gamma$ is a real $p\times dN$ matrix, $\Gamma_{N}\in\mathcal{M}_{p\times dN}(\mathbb{R})$. Since $0\le|c(x)|\le b$ for every $x\in\mathbb{R}^{dN}$ (claim 1 of Properties of the Absolute Value in an Ordered Field), and $\mathbb{R}^{dN}$ contains its origin, $0\le b$, so $c$ is bounded with bound $b$; it is uniformly continuous by hypothesis. Hence Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral, applied with $m=dN$, this $c$ and this $b$, shows that $c$ is Borel and integrable with respect to every $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, and that the function

is uniformly continuous for $W_{2}$ and the metric of The Absolute Value Metric on the Real Line, with $|G(P)|\le b$ for every $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$. In particular $c$ is bounded and Borel, as the data of The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space require, and $(\mathcal{D}_{N},\mathcal{D}_{N,\Sigma},\mathcal{E}_{N},\Sigma_{N})$ is the Langevin free-energy pair with potential $V_{N}$ and noise intensity $\sigma$ formed at the configuration level (N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level), as named in that equation.

Step 2 (identification of the solutions). By The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space §equation, the viscosity solutions, subsolutions and supersolutions of the lifted $N$-particle Hamilton-Jacobi equation with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, control cost $\theta$, common-noise matrix $\Gamma$ and running cost $c$ are exactly the functions on $\mathcal{D}_{N}$ that are viscosity solutions, subsolutions and supersolutions of The Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics in a Confining Potential on the Wasserstein Space §equation, at the configuration level (N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level), with potential $V_{N}$, noise intensity $\sigma$, discount $\lambda_{0}$, common-noise matrix $\Gamma_{N}$, control cost $\theta$ and running cost $G$ (the running cost named in The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space §operator). These are the viscosity solutions, subsolutions and supersolutions, functions on the domain $\mathcal{D}_{N}$ of the Langevin free-energy pair with potential $V_{N}$ and noise intensity $\sigma$, to which Well-Posedness of the Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics: Existence and Uniqueness of a Bounded Viscosity Solution refers when it is read at the configuration level with these data.

Step 3 (the hypotheses of the well-posedness theorem). We apply Well-Posedness of the Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics: Existence and Uniqueness of a Bounded Viscosity Solution at the configuration level (N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level), that is with $dN$ in place of $d$, with the confining potential $V_{N}$ on $\mathbb{R}^{dN}$ (Step 1), the positive numbers $\lambda_{0}$ and $\sigma$, the number $\theta$ with $0<\theta\le1$, the natural number $p$, the matrix $\Gamma_{N}\in\mathcal{M}_{p\times dN}(\mathbb{R})$ (Step 1), the running cost $g=G$, uniformly continuous for $W_{2}$ and the metric of The Absolute Value Metric on the Real Line (Step 1), and the number $b$, which satisfies $|G(P)|\le b$ for every $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ (Step 1). Its pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is then $(\mathcal{D}_{N},\mathcal{D}_{N,\Sigma},\mathcal{E}_{N},\Sigma_{N})$, its multiplicative inverse $\lambda_{0}^{-1}$ is ours, uniform continuity on a subset of $\mathcal{D}_{N}$ refers there, as here, to $W_{2}$ on $\mathcal{P}_{2}(\mathbb{R}^{dN})$ restricted to that subset, and by Step 2 its viscosity solutions, subsolutions and supersolutions are those of the statement.

Step 4 (the three clauses). Clause 1 follows from Well-Posedness of the Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics: Existence and Uniqueness of a Bounded Viscosity Solution §comparison applied as in Step 3 to $u$ and $v$: a viscosity subsolution bounded above and a viscosity supersolution bounded below of the lifted equation are, by Step 2, such functions for the equation of that theorem, so $u(P)\le v(P)$ for every $P\in\mathcal{D}_{N}$. Clause 2 is Well-Posedness of the Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics: Existence and Uniqueness of a Bounded Viscosity Solution §existence applied as in Step 3: it provides a viscosity solution $u:\mathcal{D}_{N}\to\mathbb{R}$ of the equation of that theorem, hence of the lifted equation by Step 2, with $-\lambda_{0}^{-1}b\le u(P)\le\lambda_{0}^{-1}b$ for every $P\in\mathcal{D}_{N}$. Clause 3 is Well-Posedness of the Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics: Existence and Uniqueness of a Bounded Viscosity Solution §uniqueness applied as in Step 3: bounded viscosity solutions of the lifted equation are, by Step 2, bounded viscosity solutions of the equation of that theorem, so any two of them are equal; and for $a\in\mathbb{R}$, that clause read with the real number $a$ in place of the level written $c$ there shows that a bounded viscosity solution is uniformly continuous on $\{P\in\mathcal{D}_{N}:\mathcal{E}_{N}(P)\le a\}$.