Reflected Proximal Dynamics for Accelerated High-Accuracy Log-Concave Sampling

We study sampling from $\pi (x)\propto {\mathrm{e}}^{-V(x)}$ on ${\mathbb{R}}^d$, where $V$ is $\alpha$-strongly convex and $\beta$-smooth with condition number $\kappa =\beta \mathrm{/}\alpha$. For this problem, the proximal sampler preserves the target exactly and recovers overdamped Langevin dynamics in the small-step limit, but its Gaussian resampling step discards directional information. Underdamped Langevin dynamics retains such information through momentum to accelerate exploration, motivating a proximal construction that combines target preservation with directional persistence. We introduce $\mathit{\text{reflected}}\mathit{\text{proximal}}\mathit{\text{dynamics}}$, which combines proximal sampling with an exact reflection of the auxiliary variable, conditional kinetic motion, and controlled refreshment. We develop two realizations: a flux-resampled underdamped half-turn and a persistent-momentum quarter-turn. From a warm start with constant ${\chi }^2$ divergence, both methods achieve total variation error $\epsilon$ using $\tilde{O}(\sqrt{\kappa }{d}^{1\mathrm{/}4})$ expected oracle queries. We further remove the supplied warm-start assumption by combining either sampler with existing posterior samplers in a proximal initialization stage with an optimized coarse step size. Accounting for both initialization and sampling yields $\tilde{O}({\kappa }^{6\mathrm{/}7}{d}^{1\mathrm{/}4})$ expected oracle queries. In both regimes, the dependence on the target accuracy $\epsilon$ is $\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{l}\mathrm{o}\mathrm{g}(1\mathrm{/}\epsilon )$, making reflected proximal dynamics a promising framework for accelerated high-accuracy log-concave sampling.