Abstract

Diffusion-based driving planners often work in two stages: first pick one of $K$ candidate anchor trajectories, then refine it with a denoising bridge. The refinement noise is Gaussian, and Gaussian noise has a structural weakness: large corrections through the noise are exponentially unlikely, so whatever recovery from a poor anchor the planner achieves must come from its learned denoiser alone. Heavy-tailed noise adds a second channel, occasional large jumps. We show that how the heavy tail is introduced matters more than the tail itself. If it comes from one random scale drawn per rollout, we prove that the deterministic part of the sampler is exactly the Gaussian policy: the tail is a single bet, placed before refinement begins and never revised. If instead a scale is drawn at every denoising step, the sampler can make a large correction at any point along the way. We construct this per-step variant exactly: positive $\alpha/2$-stable mixing is the only choice consistent with the bridge, and it yields a forward noising chain whose law is precisely that of an $\alpha$-stable Lévy bridge on any time grid, with a tempered finite-variance version, a one-parameter calibration, and all standard bridge machinery (training, fast samplers, warm starts) carried over unchanged to the learned planner. Experiments on NAVSIM support the theory, and they say what the per-step tail buys: anchor-independence. Trained to convergence and replicated over two training seeds, the per-step Lévy planner scores the same whichever anchor it is handed, from the best-ranked to the worst of twenty, at no cost at the deployed argmax and at unchanged compute. The margin is small because the room is small: the Gaussian bridge already recovers most of a poor anchor through its denoiser and loses only a tenth of a point from best anchor to worst in each seed, and the per-step tail closes all of it, keeping a modest but significant edge at the worst anchor (pooled paired bootstrap +0.16, 95% CI [+0.03, +0.29]). The same stress test shows how a per-rollout tail fails. In a controlled comparison of eleven variants, the scene-amortized per-rollout policy matches every standard metric yet loses 34 PDMS points when forced onto the worst-ranked anchor, while its fixed-scale counterparts simply track the Gaussian baseline, as the cancellation permits. This failure is invisible to the standard benchmark; our stress test detects it and reproduces it on the held-out 12,146-scene split.