# 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!!

• The wave operator e.g. does not generate an operator semigroup, so I believe that this is not a well-posed problem. – Matthias Ludewig Mar 17 '15 at 22:32
• Thanks! I was wondering if that might be the case, since in the scenario I proposed here, the existence of null geodesics in semi-Riemannian manifold would imply there are entire spatial directions from $x$ in which all points lie at zero geodesic distance from $x$. That seems problematic for the existence of a kernel which is supposed to be analogous to that in the Riemannian case dominated by $e^{-d(x,y)^2/ct}$. – Idempotent Mar 18 '15 at 0:50
• if we lived in a space which was only semi-Riemannian Huh? 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. – Ben Crowell May 18 '15 at 16:43
• @BenCrowell - Oh, I'm sorry, I should have been clearer - my proposed alternate scenario is a spacetime $M \approx \mathbb{R} \times \Sigma$ whose spatial slice $\Sigma$ has a semi-Riemannian metric as opposed to the usual Riemannian one. – Idempotent May 19 '15 at 13:54

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.

• Wow, thank you! I'm looking these up right now! – Idempotent May 19 '15 at 13:56

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.