Approximate Solutions to Second Order Parabolic Equations I: analytic estimates

Abstract

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 $W_a^{s,p}({\mathbb{R}}^N)$ that appear in practice.

42 pages

References in corpus (2)

• Heat kernel expansion: user's manual
• On the geometry of Riemannian manifolds with a Lie structure at infinity

Cited by in corpus (1)

• Short-time expansion of one-dimensional Fokker-Planck equations with heterogeneous diffusion