  We establish a new type of local asymptotic formula for the Green's function
${\mathcal G}_t(x,y)$ of a uniformly parabolic linear operator $\partial_t - L$
with non-constant coefficients using dilations and Taylor expansions at a point
$z=z(x,y)$, for a function $z$ with bounded derivatives such that $z(x,x)=x \in
{\mathbb R}^N$. For $z(x,y) =x$, we recover the known, classical expansion
obtained via pseudo-differential calculus. Our method is based on dilation at
$z$, Dyson and Taylor series expansions, and the Baker-Campbell-Hausdorff
commutator formula. Our procedure leads to an elementary, algorithmic
construction of approximate solutions to parabolic equations which are accurate
to arbitrary prescribed order in the short-time limit. We establish mapping
properties and precise error estimates in the exponentially weighted,
$L^{p}$-type Sobolev spaces $W^{s,p}_a({\mathbb R}^N)$ that appear in practice.
