General Mathematics

2609 Submissions

[2] ai.viXra.org:2609.0050 [pdf] replaced on 2026-10-08 15:20:46

Mean Square Error Analysis of Stochastic Runge—Kutta Integrators

Authors: Xuda Ye
Comments: 57 Pages.

We analyze the mean square error of stochastic Runge--Kutta integrators for overdamped Langevin dynamics whose potential is strongly convex outside a bounded region. A decomposition splits the local error into a mean-zero term and a smaller remainder, and the discrete Poisson equation turns their moments into a bound on the error of a time average. We carry this out for two stochastic Runge--Kutta integrators proposed by Yang & Wang (2026), both of strong order $frac32$ and weak order $2$, which evaluate the gradient of the potential and no higher derivative. We show that their laws approach the law of the exact solution at second order in Wasserstein-1 distance, up to a logarithm, uniformly in the number of steps. For a test function with bounded derivatives up to third order, we prove that the mean square error over $N$ steps with step size $h$ is $mathcal O ( frac{1}{Nh} + h^4 )$, which is the optimal order in the discretization. Experiments measure the strong and weak orders and the sampling bias on a nonconvex potential in $mathbb R^2$, and compare the integrators on a diffusion model of CIFAR-10.
Category: General Mathematics

[1] ai.viXra.org:2609.0049 [pdf] submitted on 2026-09-22 15:22:47

A Sharp Uniform-in-Time Wasserstein-1 Bound for Stochastic Runge—Kutta Integrators of Langevin Dynamics

Authors: Yuliang Wang, Xuda Ye
Comments: 23 Pages.

We consider two stochastic Runge--Kutta integrators proposed by Yang & Wang (2026) for overdamped Langevin dynamics whose potential is strongly convex outside a bounded region. In the companion paper (Ye, 2026) the law of the numerical solution was shown to approach the law of the exact solution at second order in Wasserstein-1 distance, uniformly in the number of steps, up to a logarithm of the step size. We remove the logarithm, and a Gaussian example shows that the second order is attained by the bias of the invariant law of the integrators, and that the squared bias of their time averages attains the order $h^4$ of the mean square error bound of the companion paper.
Category: General Mathematics