The Langevin Hamilton-Jacobi Equation with Common Noise and a Density Cost on the Wasserstein Space

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation. Let $V$ be a confining potential on $\mathbb{R}^{d}$, with gradient map $\nabla V$, let $\lambda_{0},\sigma,\theta\in\mathbb{R}$ be positive, let $\kappa,L\in\mathbb{R}$ be nonnegative, let $g:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$, let $\Phi$ be a convex Lipschitz integrand with constant $L$, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Langevin free-energy pair with potential $V$ and noise intensity $\sigma$, so that $\Sigma(\nu)=\nabla V+\tfrac{\sigma^{2}}{2}\xi_{\nu}$ for $\nu\in\mathcal{D}_{\Sigma}$, with $\xi_{\nu}$ the score. Every $\nu\in\mathcal{D}_{\Sigma}$ has finite entropy, hence is absolutely continuous by Basic Properties of the Entropy on the Wasserstein Space: Comparison with the Gaussian Relative Entropy, Lower Bound, Translation Invariance, Absolute Continuity, Closed Sublevel Sets and Lower Semicontinuity §absolutely-continuous, so its density cost $\mathcal{G}_{\Phi}(\nu)$ is a real number. For $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $\langle\cdot,\cdot\rangle_{\nu}$ and $\lVert\cdot\rVert_{\nu}$ are the inner product and norm of $L^{2}(\nu;\mathbb{R}^{d})$, and $\mathrm{tr}\,Y$ is the trace of $Y\in\mathcal{S}(d)$. 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 operator)¶ The Langevin Hamilton-Jacobi operator with common noise and density cost, with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, common-noise intensity $\kappa$, control cost $\theta$, running cost $g$ and integrand $\Phi$, is the function on $\mathcal{V}(\mathcal{D}_{\Sigma})\times\mathbb{R}\times\mathcal{S}(d)$ whose value at $(\nu,r,q,Y)$ is the value there of the Langevin Hamilton-Jacobi operator with common noise with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, common-noise intensity $\kappa$, control cost $\theta$ and running cost $g$, minus $\mathcal{G}_{\Phi}(\nu)$:

It is a real-valued function on $\mathcal{V}(\mathcal{D}_{\Sigma})\times\mathbb{R}\times\mathcal{S}(d)$, hence a second-order equation operator over $\mathcal{D}_{\Sigma}$.

2. (The equation)¶ The Langevin Hamilton-Jacobi equation with common noise and density cost 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 $Y\in\mathcal{S}(d)$. A classical solution, subsolution or supersolution of the equation is a classical solution, subsolution or supersolution of $F$ on $\mathcal{D}_{\Sigma}$, and a viscosity solution, subsolution or supersolution of the equation is a function $u:\mathcal{D}\to\mathbb{R}$ that is a viscosity solution, subsolution or supersolution of $F$ relative to the pair.