On collisions of Brownian particles

Ichiba, Tomoyuki; Karatzas, Ioannis

doi:10.1214/09-aap641

Cited by 52 publications

(73 citation statements)

References 26 publications

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“…Theorem 3.3. Consider parameters (g n ) n≥1 and (σ 2 n ) n≥1 which satisfy (17) g := sup n≥1 |g n | < ∞, and…”

Section: Existence and Uniqueness Results For Infinite Systemsmentioning

confidence: 99%

“…Consider any infinite classical system X = (X i ) i≥1 , of competing Brownian particles with parameters (g n ) n≥1 , (σ 2 n ) n≥1 , satisfying the condition (17). Assume the initial condition X(0) = x satisfies (5).…”

Section: Existence and Uniqueness Results For Infinite Systemsmentioning

confidence: 99%

“…For finite systems of competing Brownian particles (both classical and ranked), the question of a.s. absence of triple collisions was studied in [17,18,22]. A necessary and sufficient condition for a.s. absence of any triple collisions was found in [32]; see also [5] for related work.…”

Section: Triple Collisions For Infinite Systemsmentioning

confidence: 99%

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Infinite systems of competing Brownian particles

Sarantsev¹

2017

Ann. Inst. H. Poincaré Probab. Statist.

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Abstract. Consider a system of infinitely many Brownian particles on the real line. At any moment, these particles can be ranked from the bottom upward. Each particle moves as a Brownian motion with drift and diffusion coefficients depending on its current rank. The gaps between consecutive particles form the (infinite-dimensional) gap process. We find a stationary distribution for the gap process. We also show that if the initial value of the gap process is stochastically larger than this stationary distribution, this process converges back to this distribution as time goes to infinity. This continues the work by Pal and Pitman (2008). Also, this includes infinite systems with asymmetric collisions, similar to the finite ones from Karatzas, Pal and Shkolnikov (2016).

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“…Theorem 3.3. Consider parameters (g n ) n≥1 and (σ 2 n ) n≥1 which satisfy (17) g := sup n≥1 |g n | < ∞, and…”

Section: Existence and Uniqueness Results For Infinite Systemsmentioning

confidence: 99%

Section: Existence and Uniqueness Results For Infinite Systemsmentioning

confidence: 99%

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Infinite systems of competing Brownian particles

Sarantsev¹

2017

Ann. Inst. H. Poincaré Probab. Statist.

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“…, (σ n,n ) 2 fails to be concave [51]; but it does not seem to be known whether the strong solution continues to exist after the collision. We refer to [25,29,8] for an in-depth study of multiple collisions.…”

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confidence: 99%

Long time behaviour and mean-field limit of Atlas models

Reygner

2017

ESAIM: Procs

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Abstract. This article reviews a few basic features of systems of one-dimensional diffusions with rankbased characteristics. Such systems arise in particular in the modelling of financial markets, where they go by the name of Atlas models. We mostly describe their long time and large scale behaviour, and lay a particular emphasis on the case of mean-field interactions. We finally present an application of the reviewed results to the modelling of capital distribution in systems with a large number of agents.Résumé. Cet article présente quelques propriétés basiques des systèmes de diffusions en une dimension, interagissant à travers leur rang. De tels systèmes apparaissent en particulier dans la modélisation des marchés financiers, où ils portent le nom de modèles d'Atlas. Nous décrivons principalement leur comportement en temps long et à grande échelle, et nous nous intéressons particulièrement au cas d'interactions de champ moyen. Nous décrivons enfin une application des résultats exposés à la modélisation de la distribution du capital dans des systèmes avec un grand nombre d'agents.

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“…sample path properties of this model have undergone a detailed analysis in the case that the number of particles is fixed (see [6,7,19,20,21]). Moreover, concentration properties of the solution to (1.1) for large values of N have been studied in [34], and an analogous infinite particle system has been constructed and analyzed in [33].…”

Section: Introductionmentioning

confidence: 99%

Large Deviations for Diffusions Interacting Through Their Ranks

Dembo

Shkolnikov

Varadhan

et al. 2016

Comm Pure Appl Math

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Abstract. We prove a Large Deviations Principle (ldp) for systems of diffusions (particles) interacting through their ranks, when the number of particles tends to infinity. We show that the limiting particle density is given by the unique solution of the appropriate McKean-Vlasov equation and that the corresponding cumulative distribution function evolves according to a non-degenerate generalized porous medium equation with convection. The large deviations rate function is provided in explicit form. This is the first instance of a ldp for interacting diffusions, where the interaction occurs both through the drift and the diffusion coefficients and where the rate function can be given explicitly. In the course of the proof, we obtain new regularity results for tilted versions of such generalized porous medium equation.

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On collisions of Brownian particles

Cited by 52 publications

References 26 publications

Infinite systems of competing Brownian particles

Infinite systems of competing Brownian particles

Long time behaviour and mean-field limit of Atlas models

Large Deviations for Diffusions Interacting Through Their Ranks

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