How Does the Propagator Relate to the Heat Kernel on a Manifold?

[Orbb]

Hi,

I have a question about the relation between the propagator of a scalar field and the heat kernel. I'm not sure wether I should rather put this question into the math section: Given a Laplacian D on some manifold M, what I mean by heat kernel is just

$$K(x,y;s) = \langle x | \exp(-sD) | y \rangle$$

where x, y are distinct points on M and s is the diffusion time (in the sense that K obeys the heat eqn.). Now the propagator of a scalar field can be determined from K via

$$D^{-1}(x,y) = \int_0^{\infty} ds K(x,y;s).$$

What I want to ask now is wether there is a way to invert this expression such that given some propagator, I can determine the corresponding heat kernel. Can anyobdy help?

Cheers,
O

 

[Orbb]

Really no one with a hint? Is the question somehow ill-posed? Or should I add more detail about the specific calculation I am attempting?

 

[TriTertButoxy]

Just a guess here:
The propagator is the inverse of the operator D. So if you were given the propagator, and invert it, you would get back the operator. From there, you should be able to construct your heat kernel.