Improve the exponent in the lower-bound variational inequality

Prove that the exponent p-1 in the lower-bound variational inequality for the stochastic wave equation series expansion can be improved to p, with the aim of obtaining the optimal lower bound for the large-t asymptotics of its p-th moments.

Background

The paper discusses a lower-bound method for moment asymptotics based on Hölder’s inequality and a variational estimate applied to the series expansion of the stochastic wave equation solution. An earlier result cited in the paper contains an exponent p-1 in the relevant lower-bound estimate.

The authors note that replacing p-1 by p would yield the optimal large-time lower-bound asymptotics and explicitly formulate this as a conjecture. The present paper obtains the optimal bound through a different Laplace-transform argument, but the improvement of the earlier variational inequality itself remains unresolved.

References

In light of Proposition~4.1 with $\rho=0$, we conjecture that this exponent can be improved from $p-1$ to $p$.

— Stochastic fractional diffusion equations with spatial Gaussian noise in the Stratonovich regime  (2609.04826 - Guo et al., 4 Sep 2026) in Remark 5.3 (Remark~\ref{rem:lower}), Section 5