Fixed step size and friction cannot guarantee a uniform ballistic cold-start total-variation mixing rate for OBABO across the full class of smooth strongly convex targets, according to an arXiv preprint. Total variation measures the gap between the chain's current distribution and its own long-run, or invariant, distribution. Here, ballistic means the faster-than-diffusive rate examined by the analysis. The result is a theorem-level worst-case statement under the study's assumptions.
The tuning under examination is fixed in a strict sense. It is independent of the particular potential being sampled, the chain's current state, the iteration number and the noise realized during the run. The question is whether step size and friction chosen only from curvature bounds and dimension can work uniformly for every member of the target class. The cold starts in the analysis are deterministic point-mass starts.
One tuning, an entire target class
The target family is a normalized smooth strongly convex class. At every point, the eigenvalues of the Hessian, the matrix that records second derivatives, stay between prescribed lower and upper curvature limits. The condition number records the spread between those limits. Any guarantee being tested must hold across the full family, making worst-case behavior central to the analysis.
For OBABO, eliminating the velocity gives an exact noisy heavy-ball recursion for the positions. The central theorem uses that reduction to establish a quadratic-versus-cycling dichotomy for numerically stable tunings. One branch presents a quadratic obstruction; the other identifies a smooth strongly convex potential that produces cycling.
The cycling mechanism does not require high dimension. One explicit smooth strongly convex example is one-dimensional, has condition number 25, and uses the tuning that is optimal over quadratic targets with curvature in the interval [1, 25].
When stability gives way to cycling
In the cycling branch, dilation preserves the curvature bounds while producing a metastability window that grows exponentially with the dilation scale. The starting point used in the construction has a Wasserstein distance from equilibrium that grows only linearly with that same scale.
During this window, at least one selected deterministic cold start remains at total-variation distance of at least 3/8 from the chain's invariant law. That means the construction maintains a clear separation from stationarity throughout the stated time interval.
The lower bound concerns convergence to the invariant distribution of the discretized chain itself. It is not a direct estimate of the discretization's bias relative to the desired continuous Langevin target measure.
The lower bound has an upper-bound counterpart
The negative result comes with a complementary upper bound for OBABO. Under an explicit numerically stable fixed-parameter tuning, the chain reaches its own invariant law from a cold start on the diffusive condition-number scale, up to logarithmic factors. The proof combines Wasserstein contraction with regularization that converts Wasserstein control into total-variation control.
The two results place the proved worst-case scale between a lower bound that excludes convergence faster than diffusive order, including ballistic order, and an explicit upper bound that is diffusive apart from logarithmic factors. A further sharpness corollary rules out uniform fixed-parameter cold-start exponential bounds with any sublinear condition-number time scale under the stated polynomial-prefactor form.
Where the result stops
The conclusion is not unique to OBABO. The same lower-bound dichotomy and cold-start conclusions apply to BAOAB and the other four standard Strang splittings, although their eliminated noise structures differ.
The paper also considers the left-endpoint exponential integrator. For that method, every fixed step-size and friction choice has one of two Gaussian outcomes: it is unstable on a Gaussian, or it admits a Gaussian counterexample. This is a Gaussian alternative, not the same cycling construction established for the standard Strang splittings.
The analysis distinguishes these finite-step effects from continuous dynamics. The constructed metastable cycles have no counterpart in the continuous diffusion.
These results address unadjusted splitting chains, deterministic point-mass starts and parameters fixed independently of the target and the run. They do not settle what happens with warm starts, adaptive or target-dependent tuning, time-dependent splittings, randomized internal stages, Metropolized methods or other modified dynamics.
The hardest examples in the analysis are Gaussian benchmarks or specially constructed smooth strongly convex potentials. The supplied results do not establish how representative those examples are of practical targets.
An open boundary
The document is identified as an arXiv version-one preprint. Its conclusion is a defined result about fixed-parameter, deterministic cold-start behavior for OBABO and the standard Strang splittings, rather than a verdict on every kinetic-Langevin method.
The analysis leaves open whether changing the initialization, the way parameters are chosen or the integrator itself can permit acceleration while avoiding the cycling and instability alternatives described in the study.
Paper data and sources
Original title: Provable Non-Acceleration of Standard Strang Splittings of Kinetic Langevin Dynamics
Authors: Nawaf Bou-Rabee
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text