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

TL;DR

This paper investigates the regularizing effects of high-order and fractional PDEs driven by space-time white noise, revealing maximal spatial regularity in the fourth-order case and enhanced temporal regularity with increasing Laplacian order.

Contribution

It introduces new stochastic integral equations driven by space-time white noise, demonstrating maximal spatial regularity in the bi-Laplacian case and the impact of higher-order operators on temporal regularity.

Findings

Maximal spatial regularity achieved in the fourth-order bi-Laplacian case.

Increasing Laplacian order enhances temporal regularity of solutions.

Solutions can be smoother than Brownian sheet in space and time.

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 of the spatial-Laplacian order is entirely translated into additional temporal regularization of the SIE. We started this program in (Allouba 2013, Allouba 2006), 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 $\beta\searrow0$.