## Time-fractional and memoryful $Δ^{2^{k}}$ SIEs on $\Rp\times\Rd$: how far can we push white noise?    [PDF]

Hassan Allouba
High order and fractional PDEs have become prominent in theory and in modeling many phenomena. Here, we focus on the regularizing effect of a large class of memoryful high-order or time-fractional PDEs---through their fundamental solution---on stochastic integral equations (SIEs) driven by space-time white noise. Surprisingly, we show that maximum spatial regularity is achieved in the fourth-order-bi-Laplacian case; and any further increase of the spatial-Laplacian order is entirely translated into additional temporal regularization of the SIE. We started this program in \cite{Abtbmsie,Abtpspde}, where we introduced two different stochastic versions of the fourth order memoryful PDE associated with the Brownian-time Brownian motion (BTBM): (1) the BTBM SIE and (2) the BTBM SPDE, both driven by space-time white noise. Under wide conditions, we showed the existence of random field locally-H\"older solutions to the BTBM SIE with striking and unprecedented time-space H\"older exponents, in spatial dimensions $d=1,2,3$. In particular, we proved that the spatial regularity of such solutions is nearly locally Lipschitz in $d=1,2$. This gave, for the first time, an example of a space-time white noise driven equation whose solutions are smoother than the corresponding Brownian sheet in either time or space. In this paper, we introduce the $2\beta^{-1}$-order $\beta$-inverse-stable-L\'evy-time Brownian motion ($\beta$-ISLTBM) SIEs, driven by space-time white noise. We show that the BTBM SIE spatial regularity and its random field third spatial dimension limit are maximal among all $\beta$-ISLTBM SIEs. Furthermore, we show that increasing the order of the Laplacian $\beta^{-1}$ beyond the BTBM bi-Laplacian manifests entirely as increased temporal regularity of our random field solutions that asymptotically approaches the temporal regularity of the Brownian sheet as $\beta\searrow0$.
View original: http://arxiv.org/abs/1210.4939