Pseudodifferential multi-product representation of the solution operator of a parabolic equation

Abstract

By using a time slicing procedure, we represent the solution operator of a second-order parabolic pseudodifferential equation on ${\mathbb{R}}^n$ as an infinite product of zero-order pseudodifferential operators. A similar representation formula is proven for parabolic differential equations on a compact Riemannian manifold. Each operator in the multi-product is given by a simple explicit Ansatz. The proof is based on an effective use of the Weyl calculus and the Fefferman-Phong inequality.

Comm. Partial Differential Equations to appear (2009) 28 pages