Approximate solutions to second order parabolic equations. I: Analytic estimates

We establish a new type of local asymptotic formula for the Green's function G_t(x,y) of a uniformly parabolic linear operator ∂_t-L with nonconstant 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∈R^N. Our method is based on dilation at z, Dyson, and Taylor series expansions. We use the Baker-Campbell-Hausdorff commutator formula to explicitly compute the terms in the Dyson series. Our procedure leads to an explicit, elementary, algorithmic construction of approximate solutions to parabolic equations that are accurate to arbitrarity prescribed order in the short-time limit. We establish mapping properties and precise error estimates in the exponentially weighed, L^p-type Sobolev spaces W_a^s,p(R^N) that appear in practice.