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 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.

Keywords:

• Compact manifold
• Infinite product of operators
• Parabolic equation
• Pseudodifferential initial value problem
• Weyl quantization

AMS 2000 Subject Classification:

• 35K15
• 35S10
• 47G30
• 58J35
• 58J40

Acknowledgments

We wish to thank L. Robbiano for discussions on pullbacks of Weyl symbols. This work was initiated while the second author was visiting the Institute of Mathematics at the University of Tsukuba. Part of this work was done when the second author was on a research leave at Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions, CNRS UMR 7598, Paris, France. The second author was partially supported by l'Agence Nationale de la Recherche under grant ANR-07-JCJC-0139-01.