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Conjecture on the leading-order contribution to escape probability in the linear stochastic grey-zone model

Prove that, in the linear stochastic model y_{n+1} = (y_n − σ_n) μ + σ_n with σ_n uniformly distributed on [0,1], the escape probability Q∞(δ) is asymptotically dominated by G_M(δ), the probability of remaining in the grey zone up to n = M = − ln δ / ln μ, i.e., establish that G_M(δ) represents the leading contribution to Q∞(δ) as δ → ∞.

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Background

To analyze crisis-induced escapes under bounded fluctuations, the authors introduce a linear stochastic model of the grey zone (GZ) dynamics, derive an upper bound G_M(δ) for the escape probability Q∞(δ), and compare it with simulations. While numerical evidence suggests close agreement, a rigorous proof that G_M(δ) provides the leading-order term in Q∞(δ) is not provided.

They explicitly label this as a conjecture, framing a precise asymptotic relationship between the escape probability and the time to first eligibility for escape from the GZ.

References

Since the gap between the two quantities does not increase upon decreasing δ, we can conjecture that G_M, i.e. the probability to be still in the GZ when it becomes possible at all to escape on the left, represents the leading contribution to Q∞(δ) (for δ -> ∞).

— Crisis in time-dependent dynamical systems (2503.13152 - Olmi et al., 17 Mar 2025) in Scaling behavior paragraph (following the derivation of Eq. (theoryf) and discussion around Fig. 2(c))