Accelerating Langevin Dynamics with Weighted Laplacian Flow: Global Well-Posedness and Exponential Convergence

Abstract

We study an accelerated Langevin equation in which the classical overdamped Langevin drift is supplemented by a nonlocal gradient velocity determined through a target-weighted Poisson equation. The target $p$ is only assumed to be strictly positive and need not be normalized. Writing the dynamics in terms of $ω=\log(p/q)$ yields a coupled parabolic-elliptic system whose parabolic component contains the usual Langevin diffusion, whereas the elliptic component generates a global transport. The system is invariant under multiplication of $p$ by a positive constant. On a bounded smooth domain with homogeneous Neumann boundary conditions, we establish global existence and uniqueness of classical solutions for strictly positive $C^{2+α}$ target weights and compatible $C^{2+α}$ initial data. For long-time behavior, the oscillation of $ω$ decays at the explicit rate $e^{-t}$ and the solution density exponentially converges to target density. Moreover, an exact entropy-dissipation identity holds relative to the target density, in which the relative Fisher information and the Jeffreys divergence give the two dissipation mechanisms associated with the Langevin and weighted-Poisson components. In particular, KL divergence also converges to zero with rate $e^{-t}$ without a convexity assumption on the target.