Simulating a rare molecular transition — a protein folding, a chemical reaction — requires sampling paths that the system almost never takes. Coarse models that reduce the system to a few reaction coordinates can find these paths cheaply, but they live in the wrong space. A path through two collective variables isn't a physical trajectory through ten thousand atomic coordinates. The coarse path is a sketch. The question is whether you can turn the sketch into a painting.
Schütte, Sikorski, Kresse, and Weber (arXiv:2603.23266) show how. They take a coarse reference path and use it as guidance for the full-dimensional dynamics — a gentle force that steers the high-dimensional trajectory to track the low-dimensional reference. The bias this introduces is then corrected via Girsanov reweighting, producing an importance-sampling approximation to the true transition path ensemble.
The connection to stochastic optimal control makes the correction principled: the guided dynamics solve a control problem, and the Girsanov weight measures how much the control distorted the natural dynamics. The result is a weighted ensemble of full-dimensional trajectories that are dynamically consistent — each one is a legitimate path the real system could take, weighted by how likely it is relative to the guidance.
The structural insight is about the relationship between dimensionality and feasibility. The coarse model can find the path because it operates in a tractable space. The full model can't find the path because the space is too large. But the coarse model can't validate the path because it lacks the dimensions. The lifting procedure bridges this: the sketch provides feasibility, the full dynamics provide validity, and the reweighting makes the combination exact.