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Abstract In many situations, the combined effect of advection and diffusion greatly increases the rate of convergence to equilibrium—a phenomenon known asenhanced dissipation. Here we study the situation where the advecting velocity field generates a random dynamical system satisfying certainHarris conditions. Ifκdenotes the strength of the diffusion, then we show that with probability at least

1 - o (κ^{N})

enhanced dissipation occurs on time scales of order

| \ln κ |

, a bound which is known to be optimal. Moreover, on long time scales, we show that the rate of convergence to equilibrium is almost surelyindependentof diffusivity. As a consequence we obtain enhanced dissipation for the randomly shifted alternating shears introduced by Pierrehumbert’94.

Speeding up Langevin Dynamics by Mixing

https://doi.org/10.1137/24M168115X

Christie, Alexander; Feng, Yuanyuan; Iyer, Gautam; Novikov, Alexei (December 2025, Multiscale Modeling & Simulation)

We study an overdamped Langevin equation on the $$d$$-dimensional torus with stationary distribution proportional to $$p = e^{-U / \kappa}$$. When $$U$$ has multiple wells the mixing time of the associated process is exponentially large (of size $$e^{O(1/\kappa)}$$). We add a drift to the Langevin dynamics (without changing the stationary distribution) and obtain quantitative estimates on the mixing time. Our main result shows that the mixing time of the Langevin system can be made arbitrarily small by adding a drift that is sufficiently mixing. We provide one construction of a mixing drift, and our main result can be applied by using this drift with a large amplitude. For numerical purposes, it is useful to keep the size of the imposed drift small, and we show that the smallest allowable amplitude ensures that the mixing time is $$O( d/\kappa^2)$$, which is an order of magnitude smaller than $$e^{O(1/\kappa)}$$.

Free, publicly-accessible full text available December 31, 2026

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