  We delve deeper into the compelling regularizing effect of the Brownian-time
Brownian motion density, $\KBtxy$, on the space-time-white-noise-driven
stochastic integral equation we call BTBM SIE, which we recently introduced. In
sharp contrast to second order heat-based SPDEs--whose real-valued mild
solutions are confined to $d=1$--we prove the existence of solutions to the
BTBM SIE in $d=1,2,3$ with dimension-dependent and striking Holder regularity,
under both less than Lipschitz and Lipschitz conditions. In space, we show an
unprecedented nearly local Lipschitz regularity for $d=1,2$--roughly, the SIE
is spatially twice as regular as the Brownian sheet in these dimensions--and
nearly local H\"older 1/2 regularity in d=3. In time, our solutions are locally
H\"older continuous with exponent $\gamma\in(0,(4-d)/(8))$ for $1\le d\le3$. To
investigate our SIE, we (a) introduce the Brownian-time random walk and we use
it to formulate the spatial lattice version of the BTBM SIE; and (b) develop a
delicate variant of Stroock-Varadhan martingale approach, the K-martingale
approach, tailor-made for a wide variety of kernel SIEs including BTBM SIEs and
the mild forms of many SPDEs of different orders on the lattice. Here,
solutions types to our SIE are both direct and limits of their lattice version.
The BTBM SIE is intimately connected to intriguing fourth order SPDEs in two
ways. First, we show that it is connected to the diagonals of a new
unconventional fourth order SPDE we call parametrized BTBM SPDE. Second,
replacing $\KBtxy$ by the intimately connected kernel of our recently
introduced imaginary-Brownian time-Brownian-angle process (IBTBAP), our SIE
becomes the mild form of a Kuramoto-Sivashinsky SPDE with linear PDE part.
Ideas developed here are adapted in separate papers to give a new approach, via
our explicit IBTBAP representation, to many KS-type SPDEs in multi spatial
dimensions.
