Genericity of the $(2H+1)$-Weibull lower tail for noncommutative rough systems

Establish whether the matching $(2H+1)$-Weibull lower-tail estimate is a generic property of uniformly elliptic, noncommutative stochastic differential systems driven by fractional Brownian motion in the rough regime $H\in(1/4,1/2)$.

Background

For stochastic differential equations driven by fractional Brownian motion with $H\in(1/4,1/2)$, the Cass–Litterer–Lyons estimate provides a general $(2H+1)$-Weibull upper-tail bound. A previous line-integral example showed that this exponent is essentially sharp, motivating the conjecture that a matching lower bound should hold broadly for noncommutative systems.

The paper proves the conjectured lower bound under the quantitative noncommutativity Condition (NC), thereby giving only a partially affirmative result. The unresolved problem is to determine whether the estimate holds generically for the full class of uniformly elliptic, noncommutative systems described in the conjecture.