  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 $\beta\searrow0$.
