Small-time asymptotics with rough spatial noise

Establish whether the statement of the parabolic Anderson model’s small-time asymptotic Theorem 1 holds without modification when at least one spatial Hurst index satisfies H_j<1/2, the total mass perturbation has finite moments for sufficiently small positive times, and the rough-noise variance constant c is finite.

Background

The paper’s main theorem assumes all spatial Hurst indices are at least 1/2. For rough spatial noise, the covariance is more naturally represented through a spectral measure, and finiteness of the variance constant depends on the decay of the Fourier transform of the domain indicator.

The conjecture proposes that, when this constant is finite and moments exist, the expectation and variance asymptotics established under the paper’s assumptions continue to hold.

References

Suppose that H_j<1/2 for at least one j\geq1, and that Q_\kappa(t)'s moments are finite for small enough t>0. If the constant c in Equation: Rough c Constant is finite, 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 2, Section 3.2 (Finite Moments with Rough Space)

Characterize when c<\infty. Then, understand how the constant and the power of t in Equation: Variance must be modified when c=\infty.

— 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 3.2 (Finite Moments with Rough Space)

If d=3, H_0=1, and H_1=H_2=H_3=1/2, then there exist constants A, B\neq0 such that Q_\kappa(t)=\kappa2 A\,t{1/2}+\kappa4 B\,t\log(1/t)+O_{P}(t)\qquad\text{as }t\to0. More generally, if a model of the PAM requires multiple renormalization functions of decreasing order, say c_1(\epsilon),\ldots, c_n(\epsilon), then there are corresponding deterministic contributions of order t c_k(t) in the small-t asymptotics of Q_\kappa(t).

— The Parabolic Anderson Model's Total Mass at Small Times: Geometry, Fluctuations, and Renormalization  (2608.18834 - Lamarre et al., 19 Aug 2026) in Conjecture 3, Section 3.3.1 (Renormalization Thresholds and Deterministic Asymptotics)