Where does randomness lead in spacetime?

Bailleul, Ismaël; Raugi, Albert

doi:10.1051/ps:2008021

Cited by 10 publications

(17 citation statements)

References 14 publications

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“…The most frequently studied relativistic stochastic differential equations [11][12][13][14]17,18,20,21,24,[31][32][33][34]391,394] are driven by Brownian motion (Wiener) processes which couple to the momentum coordinates. It would be interesting to also consider other driving processes (e.g., Poisson or Lévy noise) and to compare with the results of the corresponding nonrelativistic equations [83,467,470].…”

Section: Discussionmentioning

confidence: 99%

“…Typical examples are processes described by Fokker-Planck equations (FPEs) or Langevin equations [11][12][13][14][15]17,18,20,21,[24][25][26][31][32][33][34][391][392][393][394][395]. Similar to the relativistic Boltzmann equation, relativistic FPEs in phase space can be used to describe non-equilibrium and relaxation phenomena in relativistic many-particle systems.…”

Section: Relativistic Markov Processes In Phase Spacementioning

confidence: 99%

“…As an alternative to non-Markovian diffusion models in spacetime, one can consider relativistic Markov processes in phase space [11][12][13][14][15][16][17][18][19][20][21][22]24,26,[31][32][33][34][220][221][222][332][333][334][335][336][337][338]. Typical examples are processes described by Fokker-Planck equations (FPEs) or Langevin equations [11][12][13][14][15]17,18,20,21,[24][25][26][31][32][33][34][391][392][393][394][395].…”

Section: Relativistic Markov Processes In Phase Spacementioning

confidence: 99%

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Relativistic Brownian motion

2009

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a b s t r a c tOver the past one hundred years, Brownian motion theory has contributed substantially to our understanding of various microscopic phenomena. Originally proposed as a phenomenological paradigm for atomistic matter interactions, the theory has since evolved into a broad and vivid research area, with an ever increasing number of applications in biology, chemistry, finance, and physics. The mathematical description of stochastic processes has led to new approaches in other fields, culminating in the path integral formulation of modern quantum theory. Stimulated by experimental progress in high energy physics and astrophysics, the unification of relativistic and stochastic concepts has re-attracted considerable interest during the past decade. Focusing on the framework of special relativity, we review, here, recent progress in the phenomenological description of relativistic diffusion processes. After a brief historical overview, we will summarize basic concepts from the Langevin theory of nonrelativistic Brownian motions and discuss relevant aspects of relativistic equilibrium thermostatistics. The introductory parts are followed by a detailed discussion of relativistic Langevin equations in phase space. We address the choice of time parameters, discretization rules, relativistic fluctuationdissipation theorems, and Lorentz transformations of stochastic differential equations. The general theory is illustrated through analytical and numerical results for the diffusion of free relativistic Brownian particles. Subsequently, we discuss how Langevin-type equations can be obtained as approximations to microscopic models. The final part of the article is dedicated to relativistic diffusion processes in Minkowski spacetime. Since the velocities of relativistic particles are bounded by the speed of light, nontrivial relativistic Markov processes in spacetime do not exist; i.e., relativistic generalizations of the nonrelativistic diffusion equation and its Gaussian solutions must necessarily be non-Markovian. We compare different proposals that were made in the literature and discuss their respective benefits and drawbacks. The review concludes with a summary of open questions, which may serve as a starting point for future investigations and extensions of the theory.

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Section: Discussionmentioning

confidence: 99%

Section: Relativistic Markov Processes In Phase Spacementioning

confidence: 99%

Section: Relativistic Markov Processes In Phase Spacementioning

confidence: 99%

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Relativistic Brownian motion

2009

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“…The asymptotic behavior of this G-valued diffusion was studied in [5]. A relativistic diffusion can be seen as a stochastic perturbation of the geodesic flow on a Lorentzian manifold.…”

Section: Relativistic Diffusionmentioning

confidence: 99%

“…Intuitively, {(g t , ξ t )} t≥0 describes the timelike trajectory of a small rigid object in Minkowski spacetime and consists in a stochastic perturbation of a geodesic trajectory by random perturbation of its velocity. This Lorentzian analogue to the Euclidian Brownian motion, was studied by Bailleul in [5] where he determined its Poisson boundary by providing a comprehensive description of asymptotic behaviour. This study was completed in [25] where the Lyapunov spectrum of its flow was described.…”

Section: Introductionmentioning

confidence: 99%

A Poincaré Cone Condition in the Poincaré Group

Tardif

2012

Potential Anal

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Dévissage of a Poisson Boundary Under Equivariance and Regularity Conditions

Angst¹,

Tardif²

2016

Lecture Notes in Mathematics

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We present a method that allows, under suitable equivariance and regularity conditions, to determine the Poisson boundary of a diffusion starting from the Poisson boundary of a sub-diffusion of the original one. We then give two examples of application of this dévissage method. Namely, we first recover the classical result that the Poisson boundary of Brownian motion on a rotationally symmetric manifolds is generated by its escape angle, and we then give an "elementary" probabilistic proof of the delicate result of [Bai08], i.e. the determination of the Poisson boundary of the relativistic Brownian motion in Minkowski space-time.arXiv:1311.4428v1 [math.PR]

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Where does randomness lead in spacetime?

Abstract: Mathematics Subject Classification. 60B99, 60J50, 60J45, 83A05.

Cited by 10 publications

References 14 publications

Relativistic Brownian motion

Relativistic Brownian motion

A Poincaré Cone Condition in the Poincaré Group

Dévissage of a Poisson Boundary Under Equivariance and Regularity Conditions

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