The N-Particle Hamilton-Jacobi Equation with Individual and Common Noise on the Configuration Space

In the settings of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level and of Second-Order Equations on Euclidean Open Sets, the latter read in the dimension $dN$, with the notation of The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space for traces, squared norms and halves; $\mathbb{R}^{dN}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. Let $V$ be a confining potential on $\mathbb{R}^{d}$, with $N$-particle potential $V_{N}$, of class $C^{2}$ on $\mathbb{R}^{dN}$ with gradient $DV_{N}$ by that clause; let $\lambda_{0},\sigma,\theta\in\mathbb{R}$ be positive, let $p\in\mathbb{N}$, let $\Gamma\in\mathcal{M}_{p\times d}(\mathbb{R})$ be a real $p\times d$ matrix whose $j$-th row $\gamma_{j}\in\mathbb{R}^{d}$ is the point with $i$-th coordinate $\Gamma_{ji}$, and let $c:\mathbb{R}^{dN}\to\mathbb{R}$. The letter $\sigma$ denotes the noise intensity; the swap map written $\sigma$ in Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §swap is not used.

1. (The common-noise matrix)¶ The $N$-particle common-noise matrix $\Gamma_{N}\in\mathcal{M}_{p\times dN}(\mathbb{R})$ of $\Gamma$ is the matrix whose entry in row $j$ and column $l$ is the $l$-th coordinate of the diagonal point $\gamma_{j}^{\oplus}\in\mathbb{R}^{dN}$, for $j\in[p]$ and $l\in[dN]$; its $j$-th row is thus $\gamma_{j}^{\oplus}$, and by The Trace as a Sum of Quadratic Forms, its Monotonicity and a Norm Bound §rows

the product $\Gamma_{N}^{\top}\Gamma_{N}X$ being unambiguous by Associativity of the Matrix Product.

2. (The operator)¶ The $N$-particle Hamilton-Jacobi operator with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, control cost $\theta$, common-noise matrix $\Gamma$ and running cost $c$ is the second-order equation operator on $\mathbb{R}^{dN}$

for $x\in\mathbb{R}^{dN}$, $r\in\mathbb{R}$, $\zeta\in\mathbb{R}^{dN}$ and $X\in\mathcal{S}(dN)$.

3. (The equation)¶ The $N$-particle Hamilton-Jacobi equation is

that is, $F(x,u(x),Du(x),D^{2}u(x))=0$ for $x\in\mathbb{R}^{dN}$. Its classical sub- and supersolutions are those of Classical Subsolution and Supersolution of a Second-Order Equation and its viscosity sub- and supersolutions those of Viscosity Subsolution and Supersolution of a Second-Order Equation, for the operator $F$ on $\mathbb{R}^{dN}$.