Strong solutions of stochastic equations with rank-based coefficients

Ichiba, Tomoyuki; Karatzas, Ioannis; Shkolnikov, Mykhaylo

doi:10.1007/s00440-012-0426-3

Cited by 59 publications

(109 citation statements)

References 15 publications

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“…A more precise definition is given in Definitions 6 and 7 later in this article. This system was studied in [35,18]. For g 1 = 1, g 2 = g 3 = .…”

Section: Introductionmentioning

confidence: 99%

“…Until the first moment of a triple collision, it exists in the strong sense and is pathwise unique, [18]. It is not known whether it exists in the strong sense after this first triple collision.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, it is an easy exercise to show that a system of i.i.d standard Brownian motions starting from the same point is not rankable from bottom to top at any fixed time t > 0. Some sufficient conditions for weak existence and uniqueness in law are found in [35,18]. We restate them in Theorem 3.1 in a slightly different form: (5) lim i→∞ x i = ∞ and…”

Section: Introductionmentioning

confidence: 99%

“…Strong existence and pathwise uniqueness for finite systems are known from [18] to hold until the first triple collision, when three or more particles simltaneously occupy the same position. It is not known whether these hold after this first triple collision.…”

Section: Introductionmentioning

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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“…A more precise definition is given in Definitions 6 and 7 later in this article. This system was studied in [35,18]. For g 1 = 1, g 2 = g 3 = .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Infinite systems of competing Brownian particles

Sarantsev¹

2017

Ann. Inst. H. Poincaré Probab. Statist.

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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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Brownian Particles with Rank‐Dependent Drifts: Out‐of‐Equilibrium Behavior

Cabezas¹,

Dembo²,

Sarantsev

et al. 2019

Comm Pure Appl Math

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We study the long‐range asymptotic behavior for an out‐of‐equilibrium, countable, one‐dimensional system of Brownian particles interacting through their rank‐dependent drifts. Focusing on the semi‐infinite case, where only the leftmost particle gets a constant drift to the right, we derive and solve the corresponding one‐sided Stefan (free‐boundary) equations. Via this solution we explicitly determine the limiting particle‐density profile as well as the asymptotic trajectory of the leftmost particle. While doing so we further establish stochastic domination and convergence to equilibrium results for the vector of relative spacings among the leading particles. © 2019 Wiley Periodicals, Inc.

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Strong solutions of stochastic equations with rank-based coefficients

Cited by 59 publications

References 15 publications

Infinite systems of competing Brownian particles

Infinite systems of competing Brownian particles

Large Deviations for Diffusions Interacting Through Their Ranks

Brownian Particles with Rank‐Dependent Drifts: Out‐of‐Equilibrium Behavior

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