# Relativistic diffusion processes and random walk models

## Abstract

The nonrelativistic standard model for a continuous, one-parameter diffusion process in position space is the Wiener process. As is well known, the Gaussian transition probability density function (PDF) of this process is in conflict with special relativity, as it permits particles to propagate faster than the speed of light. A frequently considered alternative is provided by the telegraph equation, whose solutions avoid superluminal propagation speeds but suffer from singular (noncontinuous) diffusion fronts on the light cone, which are unlikely to exist for massive particles. It is therefore advisable to explore other alternatives as well. In this paper, a generalized Wiener process is proposed that is continuous, avoids superluminal propagation, and reduces to the standard Wiener process in the nonrelativistic limit. The corresponding relativistic diffusion propagator is obtained directly from the nonrelativistic Wiener propagator, by rewriting the latter in terms of an integral over actions. The resulting relativistic process is non-Markovian, in accordance with the known fact that nontrivial continuous, relativistic Markov processes in position space cannot exist. Hence, the proposed process defines a consistent relativistic diffusion model for massive particles and provides a viable alternative to the solutions of the telegraph equation.

- Authors:

- Institut fuer Physik, Universitaet Augsburg, Theoretische Physik I, Universitaetstrasse 1, D-86135 Augsburg (Germany)

- Publication Date:

- OSTI Identifier:
- 21011033

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 4; Other Information: DOI: 10.1103/PhysRevD.75.043001; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DIFFUSION; GRAPH THEORY; LIGHT CONE; MARKOV PROCESS; MATHEMATICAL SOLUTIONS; PROBABILITY; PROPAGATOR; RANDOMNESS; RELATIVISTIC RANGE; RELATIVITY THEORY; STANDARD MODEL; VISIBLE RADIATION

### Citation Formats

```
Dunkel, Joern, Talkner, Peter, and Haenggi, Peter.
```*Relativistic diffusion processes and random walk models*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVD.75.043001.

```
Dunkel, Joern, Talkner, Peter, & Haenggi, Peter.
```*Relativistic diffusion processes and random walk models*. United States. doi:10.1103/PHYSREVD.75.043001.

```
Dunkel, Joern, Talkner, Peter, and Haenggi, Peter. Thu .
"Relativistic diffusion processes and random walk models". United States.
doi:10.1103/PHYSREVD.75.043001.
```

```
@article{osti_21011033,
```

title = {Relativistic diffusion processes and random walk models},

author = {Dunkel, Joern and Talkner, Peter and Haenggi, Peter},

abstractNote = {The nonrelativistic standard model for a continuous, one-parameter diffusion process in position space is the Wiener process. As is well known, the Gaussian transition probability density function (PDF) of this process is in conflict with special relativity, as it permits particles to propagate faster than the speed of light. A frequently considered alternative is provided by the telegraph equation, whose solutions avoid superluminal propagation speeds but suffer from singular (noncontinuous) diffusion fronts on the light cone, which are unlikely to exist for massive particles. It is therefore advisable to explore other alternatives as well. In this paper, a generalized Wiener process is proposed that is continuous, avoids superluminal propagation, and reduces to the standard Wiener process in the nonrelativistic limit. The corresponding relativistic diffusion propagator is obtained directly from the nonrelativistic Wiener propagator, by rewriting the latter in terms of an integral over actions. The resulting relativistic process is non-Markovian, in accordance with the known fact that nontrivial continuous, relativistic Markov processes in position space cannot exist. Hence, the proposed process defines a consistent relativistic diffusion model for massive particles and provides a viable alternative to the solutions of the telegraph equation.},

doi = {10.1103/PHYSREVD.75.043001},

journal = {Physical Review. D, Particles Fields},

number = 4,

volume = 75,

place = {United States},

year = {Thu Feb 15 00:00:00 EST 2007},

month = {Thu Feb 15 00:00:00 EST 2007}

}