The Dyson Hamilton-Jacobi Equation with Common Noise and a Confining Potential on the Wasserstein Space of the Real Line

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, in dimension $d=1$, with the identifications of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative. Let $V$ be a confining potential, with derivative $V'$, let $\lambda_{0},\beta\in\mathbb{R}$ be positive, let $\kappa\in\mathbb{R}$ be nonnegative, let $g:\mathcal{P}_{2}(\mathbb{R})\to\mathbb{R}$, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the confined logarithmic-energy pair with potential $V$ and inverse temperature $\beta$, a penalty pair by The Confined Logarithmic-Energy Pair is a Displacement Convex, Wasserstein-Coercive Penalty Pair with Closed Score and Regular Penalised Maxima §pair, so that $\Sigma(\nu)=V'-\tfrac{\beta}{4}\Xi_{\nu}$ for $\nu\in\mathcal{D}_{\Sigma}$, with $\Xi_{\nu}$ the free score. For $\nu\in\mathcal{P}_{2}(\mathbb{R})$, $\langle\cdot,\cdot\rangle_{\nu}$ and $\lVert\cdot\rVert_{\nu}$ are the inner product and norm of $L^{2}(\nu;\mathbb{R})$, and $\mathrm{tr}\,Y$ is the trace of $Y\in\mathcal{S}(1)$.

1. (The operator)¶ The Dyson Hamilton-Jacobi operator with common noise and confining potential $V$ is the Hamilton-Jacobi operator with common noise and penalty drift of that pair with discount $\lambda_{0}$, common-noise intensity $\kappa$, control cost $1$ and running cost $g$, a second-order equation operator over $\mathcal{D}_{\Sigma}$; by the formula for $\Sigma$, its value at $(\nu,r,q,Y)$ is

2. (The equation)¶ The Dyson Hamilton-Jacobi equation with common noise and confining potential $V$ is

that is, $F(\nu,r,q,Y)=0$, with $(\nu,q)$ in the bundle $\mathcal{V}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$ and $Y\in\mathcal{S}(1)$. Its classical and viscosity solutions, subsolutions and supersolutions are those of The Discounted Hamilton-Jacobi Equation with Common Noise and a Penalty Drift on the Wasserstein Space §equation for this pair and these coefficients.