Time-fractional and memoryful $\Delta^{2^{k}}$ SIEs on $\mathbb{R}_{+}\times\mathbb{R}^{d}$: How far can we push white noise?

Abstract

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 solutions—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 in the spatial-Laplacian order is entirely translated into additional temporal regularization of the SIE. We started this program in [Discrete Contin. Dyn. Syst. Ser. A 33 (2013) 413–463, Stoch. Dyn. 6 (2006) 521–534], 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-Hölder solutions to the BTBM SIE with striking and unprecedented time-space Hölder 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-Lévy-time Brownian motion ($\beta$-ISLTBM) SIEs, $\beta\in \{1/2^{k};k\in\mathbb{N}\}$, driven by space–time white noise. Based on the dramatic regularizing effect of the BTBM density ($\beta=1/2$), and since the kernels in these $\beta$-ISLTBM SIEs are fundamental solutions to higher order Laplacian PDEs; one may suspect that we get even more dramatic spatial regularity than the BTBM SIE case. We show, however, that the BTBM SIE spatial regularity and its random field third spatial dimension limit are maximal among all $\beta$-ISLTBM SIEs—no matter how high we take the order $1/\beta$ of the Laplacian. This gives a limit as to how far we can push the SIEs spatial regularity when driven by the rough white noise. 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 that of the Brownian sheet as $\beta\searrow0$. Our solutions are both direct and lattice limit solutions. We treat many stochastic fractional PDEs and their corresponding higher order SPDEs, including BTBM and $\beta$-inverse-stable-Lévy-time Brownian motion SPDEs, in separate articles.