Tagged Particle Limit for a Fleming-Viot Type System

Grigorescu, Ilie; Kang, Min Ho

doi:10.1214/ejp.v11-316

Cited by 12 publications

(12 citation statements)

References 17 publications

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“…Moreover, since is finite, T t m(ξ ) is close to the unique QSD, uniformly in ξ , as implied by (6). These two facts imply the following, which is our main result.…”

Section: Theorem 1 ([3]) Assume That Is Finite and That The Processsupporting

confidence: 68%

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Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces

Asselah

Ferrari

Groisman

2011

Journal of Applied Probability

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Consider a continuous-time Markov process with transition rates matrix Q in the state space ∪ {0}. In the associated Fleming-Viot process N particles evolve independently in with transition rates matrix Q until one of them attempts to jump to state 0. At this moment the particle jumps to one of the positions of the other particles, chosen uniformly at random. When is finite, we show that the empirical distribution of the particles at a fixed time converges as N → ∞ to the distribution of a single particle at the same time conditioned on not touching {0}. Furthermore, the empirical profile of the unique invariant measure for the Fleming-Viot process with N particles converges as N → ∞ to the unique quasistationary distribution of the one-particle motion. A key element of the approach is to show that the two-particle correlations are of order 1/N .

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“…Moreover, since is finite, T t m(ξ ) is close to the unique QSD, uniformly in ξ , as implied by (6). These two facts imply the following, which is our main result.…”

Section: Theorem 1 ([3]) Assume That Is Finite and That The Processsupporting

confidence: 68%

“…This model and generalizations of it were studied in several papers; see, e.g. [1], [2], [5], [6], and [7], which dealt with diffusions in bounded or unbounded domains. These works had to address the serious problem of nonexplosion of the number of hits of the boundary, and this required sophisticated analysis.…”

Section: The Associated Fleming-viot Processmentioning

confidence: 99%

Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces

Asselah

Ferrari

Groisman

2011

Journal of Applied Probability

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“…It is worth mentioning that the method used can be generalized to diffusions under natural regularity conditions. Finally, this approach leads to an exact derivation of the asymptotic law of the tagged particle, together with a proof of the propagation of chaos presented in [4].…”

Section: Introductionmentioning

confidence: 99%

Hydrodynamic limit for a Fleming–Viot type system

Grigorescu

Kang

2004

Stochastic Processes and their Applications

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We consider a system of N Brownian particles evolving independently in a domain D. As soon as one particle reaches the boundary it is killed and one of the other particles is chosen uniformly and splits into two independent particles resuming a new cycle of independent Brownian motion until the next boundary hit. We prove the hydrodynamic limit for the joint law of the empirical measure process and the average number of visits to the boundary as N approaches infinity.

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“…Both models have been considered by Burdzy et al [5] and by Grigorescu and Kang [12,13]. Their relation to models in physics and probability is discussed in [4,5].…”

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confidence: 99%

A stationary Fleming–Viot type Brownian particle system

Löbus¹

2008

Math. Z.

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We consider a system {X 1 , . . . , X N } of N particles in a bounded d-dimensional domain D. During periods in which none of the particles X 1 , . . . , X N hit the boundary ∂ D, the system behaves like N independent d-dimensional Brownian motions. When one of the particles hits the boundary ∂ D, then it instantaneously jumps to the site of one of the remaining N − 1 particles with probability (N − 1) −1 . For the system {X 1 , . . . , X N }, the existence of an invariant measure ν ν has been demonstrated in Burdzy et al. [Comm Math Phys 214(3):679-703, 2000]. We provide a structural formula for this invariant measure ν ν in terms of the invariant measure m of the Markov chain ξ which returns the sites the process X := (X 1 , . . . , X N ) jumps to after hitting the boundary ∂ D N . In addition, we characterize the asymptotic behavior of the invariant measure m of ξ when N → ∞. Using the methods of the paper, we provide a rigorous proof of the fact that the stationary empirical measure processes 1 N N i=1 δ X i converge weakly as N → ∞ to a deterministic constant motion. This motion is concentrated on the probability measure whose density with respect to the Lebesgue measure is the first eigenfunction of the Dirichlet Laplacian on D. This result can be regarded as a complement to a previous one in Grigorescu and Kang [Stoch Process Appl 110(1):111-143, 2004].

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Tagged Particle Limit for a Fleming-Viot Type System

Cited by 12 publications

References 17 publications

Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces

Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces

Hydrodynamic limit for a Fleming–Viot type system

A stationary Fleming–Viot type Brownian particle system

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