Cass–Litterer–Lyons tail estimate conjecture for noncommutative systems

Let $X$ solve a multidimensional stochastic differential equation driven by fractional Brownian motion, with initial condition $X_0=x_0$ , and suppose that the system is uniformly elliptic and its diffusion vector fields are noncommutative. The conjecture is that there exist constants $c_1,c_2,R_0>0$ such that $\mathbb{P}(|X_1-x_0|>R)\ge c_1\exp(-c_2R^2)$ for every $R\ge R_0$ . The precise range of the Hurst parameter and the additional hypotheses under which this should hold remain part of the open problem.

References

Primary source

arXiv

Additional references

Optimal Tail Estimates for Differential Equations Driven by Fractional Brownian Motion — arXiv — Horatio Boedihardjo, Xi Geng, Sheng Wang

Posted 2 days ago · edited 2 days ago

Progress summary

Refreshed 2 days ago

Claimed progress

A new unrefereed preprint proves the expected tail behavior in some settings but gives counterexamples in others, so the general conjecture remains open.

The conjecture concerns Gaussian lower-tail behavior for noncommutative systems driven by fractional Brownian motion. The latest work separates regimes where the conjectured behavior holds from Young-regime cases where it fails.

October 2026 partial results

Horatio Boedihardjo, Xi Geng, and Sheng Wang claim Weibull or Gaussian lower-tail estimates in selected regimes, construct Young-regime counterexamples, and prove positive results for periodic $2\times 2$ systems. These results advance the conjecture but do not settle it in full generality.

Current status (as of October 2026): Selected regimes have claimed results, including counterexamples in the Young regime, but the general Cass–Litterer–Lyons conjecture remains open and the preprint is unverified.

Sources

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