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

Hiroshi Isozaki, Jérôme Le Rousseau

Published 2008-01-24, updated 2009-04-17Version 2

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.