Small-time asymptotics with rough temporal noise

Establish whether the statement of the parabolic Anderson model’s small-time asymptotic Theorem 1 holds without modification when the temporal Hurst index satisfies H_0<1/2, all spatial Hurst indices satisfy H_j\geq1/2, and the total mass perturbation has finite moments for sufficiently small positive times.

Background

The paper proves its principal expectation and variance asymptotics under H_0\geq1/2 and spatial Hurst indices H_j\geq1/2. The authors conjecture that the same formulas remain valid in the rough-time regime H_0<1/2 whenever the relevant moments remain finite.

If true, this extension would place the geometry–fluctuations transition in a supercritical temporal-roughness regime in which fluctuations may become the sole dominant contribution to the total mass asymptotics.

References

If H_0<1/2, H_j\geq1/2 for all j\geq1, and the moments of Q_\kappa(t) are finite for small t>0, then the statement of Theorem \ref{Theorem: Main} holds without modification.

— The Parabolic Anderson Model's Total Mass at Small Times: Geometry, Fluctuations, and Renormalization  (2608.18834 - Lamarre et al., 19 Aug 2026) in Conjecture 1, Section 3.1 (Rough Time and Fluctuations)

If \rho\leq-1/2 (i.e., d-H\geq2H_0-1/2), does there sometimes exist a scaling function s such that s(t)\big(|D|- M_\kappa(t)\big) converges in probability to a \kappa-dependent constant as t\to0?

— The Parabolic Anderson Model's Total Mass at Small Times: Geometry, Fluctuations, and Renormalization  (2608.18834 - Lamarre et al., 19 Aug 2026) in Open Problem, Section 3.3.2 (A Geometric-Renormalization Transition)