Approximate Solutions to Second Order Parabolic Equations I: analytic estimates

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We establish a new type of local asymptotic formula for the Green's function Gₜ(x,y) of a uniformly parabolic linear operator ∂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 ∈ Rⁿ. 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, Lᵖ-type Sobolev spaces Wˢ,ᵖₐ(Rⁿ) that appear in practice.