Diffusion on a semi-Riemannian manifold? 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!!
 A: This question has been addressed in some cases through a probabilistic approach. You probably know that the heat equation on a Riemannian manifold is intimately tied with its Brownian motion; one way to generalize the diffusion of heat is thus to generalize the Brownian motion; this has been studied in the case of Lorentzian signature under the name of Relativistic Diffusion.
I am not familiar with this domain, but I know someone who is: Jürgen Angst; checking his first paper's abstract, it seems that the subject was started in the following works:


*

*Debbasch et al. J. Math. Phys. 40, 2891 (2001); Eur. Phys. J. 19, 37 (2001);23, 487 (2001); J. Stat. Phys. 88, 945 (1997);90, 1179 (1998),

*Dunkel and Hänggi Phys. Rev. E 71, 016124 (2005);72, 036106 (2005) 


I guess than from there and Angst's page you should find a large number of references.
A: Thanks for asking this question and also thanks to @Benoît Kloeckner for the suggestions. I had the same question too.
In case you haven't already found the appropriate equation you are looking for, here is an one-dimensional relativistic diffusion equation [http://plms.oxfordjournals.org/content/107/6/1395].
$$
u_t = \nu \left(\frac{u u_x}{\sqrt{u^2 +(\nu^2/c^2) (u_x)^2}}\right)_x,  x \in \mathbb{R}, t >0.
$$
Since it relativistic, we are dealing with Lorentzian manifold which is a semi-Riemannian manifold. Hope this helps. 
