How to define the diffusion equation in spacetime?

Question

I would like to generalize my work on diffusion into 4D space. What's the standard way to define the diffusion equation for relativistic concerns? How does the MSD (mean square displacement) will change in the relativistic case?  

Answer 1

Standard diffusion is described in terms of a parabolic partial differential equation, while relativity is governed by hyperbolic differential equations, due to the constancy of the speed of light. Thus there is no obvious generalization.

Entry points for various ways to generalize the notion are the following papers, in reverse chronological order. (The old papers are still worth reading.)

O'Hara, P., & Rondoni, L. (2015). Brownian motion in Minkowski space. Entropy 17, 3581-3594.

Terras, A. (2013). The Poincaré Upper Half-Plane. In Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane (pp. 149-376). Springer New York.

Baeumer, B., Meerschaert, M. M., & Naber, M. (2010). Stochastic models for relativistic diffusion. Physical Review E 82, 011132.

Dunkel, J., & Hänggi, P. (2009). Relativistic Brownian motion.Physics Reports 471, 1-73. https://arxiv.org/abs/0812.1996

Haba, Z. (2009). Relativistic diffusion. Physical Review E 79, 021128.

Kostädt, P., & Liu, M. (2000). Causality and stability of the relativistic diffusion equation. Physical Review D 62, 023003.

Posilicano, A. (1997). Poincaré-invariant Markov processes and Gaussian random fields on relativistic phase space.Letters in Mathematical Physics 42, 85-93.

Dudley, R. M. (1966). Lorentz-invariant Markov processes in relativistic phase space. Arkiv för Matematik, 6, 241-268. (See also here.)

Hunt, G. A. (1956), Semi-groups of measures on Lie groups, Transactions o/the American Mathematical Society 81, 264-293.

Answer 2

May I ask what motivate the question? If it is about writing down a covariant action for a non-conservative system a good starting point is Chad Galley's papers. (e.g. https://arxiv.org/abs/1412.3082).

Comments

I want to study random walks in spacetime and study relativistic Anomalous diffusion.

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Not sure whether this is helpful, but a quick googling found the following thesis.

https://math.mit.edu/~dunkel/Diplom/diss.pdf

In general, formulation of relativistic thermodynamics with dissipation is a huge area with many ideas. In astrophysical community some of names are: Israel-Stewart, Eckart etc. You can find relevant references in the thesis linked.

Good Luck.

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You should put youir remark into the answer.