The simulation of Brownian dynamics with hydrodynamic interactions requires approximately solving stochastic differential equations with multiplicative noise. These equations bear some resemblance to the Black‐Scholes SDE from mathematical finance; except here the dynamics to be simulated is along an infinitely long trajectory. Long‐time simulations of such systems require schemes that are at once finite‐time accurate, ergodic, and fast mixing. Using the Metropolis‐Hastings algorithm, I will show how one can turn any explicit integrator that accurately represents the multiplicative noise ‐ even a "drift‐free" random walk ‐ into a long‐time stable and accurate scheme. I will provide conditions under which the method is geometrically ergodic with respect to the exact equilibrium distribution of the SDE. Moreover, by invoking a fluctuation‐dissipation theorem, I will explain why this algorithm is weakly convergent to the solution of the SDE with order 0.5, and pathwise accurate too. Another important property of this algorithm is that it does not require computing the divergence of the so‐called mobility tensor ‐ which is implicitly defined in terms of a steady‐state, inhomogeneous Stokes equation.

This talk features collaborative work with Eric Vanden‐Eijnden (NYU) and Aleksander Donev (NYU).