How Should Parallel Langevin Chains Share Noise?

Parallel Langevin samplers are usually run as $K$ independent chains and then averaged. Independence is convenient, but it is not required by the unadjusted Langevin algorithm (ULA): each chain only needs a standard Gaussian noise marginal at each step. We use this freedom by coupling the same-step noises across chains. The coupling is simple: draw $K$ iid Gaussian noises, subtract their across-chain mean, and rescale. Each chain still has the ordinary ULA law, but the noise injected into the ensemble average is exactly zero. For quadratic targets, this removes the Langevin-noise contribution from every equal-weight linear summary of the ensemble mean at every finite horizon. With deterministic or zero-sum randomized starts, these summaries have zero total variance. On UCI Bayesian logistic posteriors with $K=8$, the same construction reduces ensemble-mean trace variance to $6\times 10^{-3}$, $2\times 10^{-3}$, and $5\times 10^{-4}$ of matched iid ensembles on WDBC, Spambase, and Adult; reference MSE against a longer iid run falls by roughly $4$ to $14\times$ after accounting for error in the finite reference run. The gain is scoped: zero-sum coupling cancels linear fluctuations, while centered quadratic observables have factor $1/(K-1)$ rather than iid's $1/K$, and random starts, minibatches, and nonquadratic curvature add explicit residual variance terms. Synthetic and real-data experiments match the exact cancellation where it applies and the predicted behavior outside that regime.