Characterize existence in the subcritical fractional regime

Determine necessary and sufficient conditions for the existence of square-integrable mild Stratonovich solutions to the stochastic fractional diffusion equation when the fractional parameters satisfy b+2r<1, thereby reconciling the necessary Dalang condition with the stronger sufficient integrability condition.

Background

The paper proves that Dalang’s condition is necessary for a square-integrable mild Stratonovich solution in the considered class of stochastic fractional diffusion equations. When b+2r≥1, the paper also proves that this condition is sufficient, yielding a necessary-and-sufficient criterion.

For b+2r<1, the paper establishes only a stronger sufficient condition involving an exponent δ, while retaining Dalang’s condition as the necessary condition. Under the scaling assumption on the spatial covariance, these become different restrictions on the homogeneity parameter α, so the exact existence threshold remains unresolved.

References

It remains to consider the range $b+2r<1$, where necessary and sufficient conditions do not presently coincide. Indeed, Theorem~\ref{th:necessity-dalang} shows that Dalang's condition is necessary in this range, while the sufficient condition is e:dalang-plus.

e:dalang-plus:

∫Rd(11+∣ξ∣a)δμ(ξ)<∞.\int_{R^d}\left(\frac{1}{1+|\xi|^a}\right)^{\delta}\mu( \xi)<\infty.

— Stochastic fractional diffusion equations with spatial Gaussian noise in the Stratonovich regime  (2609.04826 - Guo et al., 4 Sep 2026) in Remark following Corollary after Theorem 2.2 (discussion of Theorem 2.1 and Theorem 2.2)