Extend the compact-support-a.e.-times result to the non-local SPDE with fractional Laplacian

Establish for non-negative, integrable solutions u to the non-local stochastic heat equation ∂_t u(t,x) = -(-Δ)^{α/2} u(t,x) + u(t,x)^γ Ẇ(t,x) on ℝ (with γ ∈ (0,1/2)) that, for every T > 0, the set of times t ∈ [0,T] for which supp(u(t,·)) is compact has full Lebesgue measure almost surely, and ascertain that the classical compact support property (boundedness of the union of supports over [0,T]) fails in this setting, analogously to the results proved here for the spatially discrete infinite-dimensional SDE system.

Background

The paper proves for a class of spatially discrete infinite-dimensional SDEs with Hölder noise coefficient σ(x)≈x^γ, γ∈(0,1/2), that the set of times at which the solution has compact support has full measure, while the classical compact support property (boundedness of the union of supports on [0,T]) fails. This sharpens the contrast with the γ=1/2 superprocess case, which exhibits instantaneous propagation of support.

The authors expect their methods, based on excursion analysis of semimartingales with a quadratic variation lower bound, to extend to non-local SPDEs driven by the fractional Laplacian, but they do not carry out this extension here and instead state it as a conjecture.