A great deal of literature exists on the heat equation and heat kernel for a Riemannian manifold. The Laplace-Beltrami operator in the given metric replaces the flat Laplacian in the heat equation, and the heat kernel becomes:

\begin{equation*} p(t,x,y) = F(t,x,y) \exp \left( - \frac{d(x,y)^2}{ct} \right) \end{equation*}

where $d(x,y)$ is the geodesic distance between $x$ and $y$ and $F$ is some function depending on the Riemannian spatial geometry, instead of the familiar:

\begin{equation*} p(t,x,y) = \left( \frac{1}{4 \pi t} \right)^{n/2} \exp \left( - \frac{\lvert x-y \rvert^2}{4t} \right) \end{equation*}

from Euclidean space.

Thus, even living in a curved space, we can still investigate how an uneven distribution of perfume would diffuse through such a spatial manifold if filled with air.

My question concerns how much we would be able to say about diffusion if we lived in a space which was only semi-Riemannian, so that the metric was nondegenerate but not positive definite. This changes things fundamentally, since of course the Laplace-Beltrami operator is no longer elliptic but hyperbolic. But one can still at least intuitively imagine spraying perfume in such a space and seeing where it goes.

Has any analog of the heat equation / heat kernel been studied for a semi-Riemannian manifold? If so, could anyone suggest search terms or a reference?

Many thanks!!

if we lived in a space which was only semi-RiemannianHuh? We do live in such a space. There is no a priori guarantee that such a space is time-orientable, has Cauchy surfaces, or lacks closed, timelike curves. You may be interested in notions from general relativity such as global hyperbolicity. See, e.g., Hawking and Ellis, The Large Scale Structure of Space-Time, p. 206. $\endgroup$ – Ben Crowell May 18 '15 at 16:43