  We use our earlier Brownian-time framework to formulate and establish global
uniqueness and local-in-time existence of the Burgers incarnation of the
Kuramoto-Sivashinsky PDE on $\mathbb{R}_+\times\mathbb{R}^d$, in the class of
time-continuous $L^{2p}$-valued solutions, $p\ge1$, for every $d<6p$. We assume
neither space compactness, nor spatial coordinates dependence, nor smallness of
initial data. The surprising discovery of the $6p$-th dimension bound, even for
local solutions, is revealed by our approach and the Brownian-time kernel --
the Brownian average of an angled $d$-dimensional Schr\"odinger propagator --
at its heart. We use this kernel to give a systematic approach, for all
dimensions simultaneously, including a novel formulation -- even in the
well-known $d=1$ case -- of the KS equation. This yields the estimates leading
to this article's conclusions. We achieve the stated results by fusing some of
our earlier Brownian-time stochastic processes constructions and ideas --
encoded in the aforementioned kernel -- with analytic ones, including complex
and harmonic analysis; by employing suitable $N$-ball approximations together
with fixed point theory; and by an adaptation of the stochastic analytic
stopping-time technique to our deterministic setting. Using a separate
strategy, that is also built on our Brownian-time paradigm, we treat the global
wellposedness of the multidimensional KS equation in a followup upcoming
article. This work also serves as a template for another forthcoming article in
which we prove similar results for the time-fractional Burgers equation in
multidimensional space.
