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

TL;DR

This paper introduces a novel representation of the solution operator for second-order parabolic pseudodifferential equations as an infinite product of simpler zero-order operators, utilizing advanced calculus techniques.

Contribution

It provides a new multi-product formula for parabolic equations on Euclidean space and manifolds, with explicit operators and rigorous proof methods.

Findings

Representation as infinite product of zero-order operators

Explicit Ansatz for each operator

Application of Weyl calculus and Fefferman-Phong inequality

Abstract

By using a time slicing procedure, we represent the solution operator of a second-order parabolic pseudodifferential equation on $\R^n$ 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.