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!!



Diffusion on a semi-Riemannian manifold?
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](http://perso.univ-rennes1.fr/jurgen.angst/recherche.htm); 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.


