Immortal particle for a catalytic branching process

Grigorescu, Ilie; Kang, Min Ho

doi:10.1007/s00440-011-0347-6

Cited by 30 publications

(43 citation statements)

References 17 publications

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“…Note that, in the latter, the authors gives a general set of sufficient assumptions for non explosion, some of them being further generalized in [12]. The upcoming result is exactly Theorem 1 of Section 2.1 in [6], in the simple case of smooth domains.…”

Section: Example: Feller Process With Hard Obstaclementioning

confidence: 94%

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A central limit theorem for Fleming–Viot particle systems

Cérou¹,

Delyon²,

Guyader³

et al. 2020

Ann. Inst. H. Poincaré Probab. Statist.

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Fleming-Viot type particle systems represent a classical way to approximate the distribution of a Markov process with killing, given that it is still alive at a final deterministic time. In this context, each particle evolves independently according to the law of the underlying Markov process until its killing, and then branches instantaneously on another randomly chosen particle. While the consistency of this algorithm in the large population limit has been recently studied in several articles, our purpose here is to prove Central Limit Theorems under very general assumptions. For this, we only suppose that the particle system does not explode in finite time, and that the jump and killing times have atomless distributions. In particular, this includes the case of elliptic diffusions with hard killing.

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Section: Example: Feller Process With Hard Obstaclementioning

confidence: 94%

“…Assumption (B) does not follow from a classical result. It is proved for instance in [6] for regular diffusions and smooth boundary. Note that, in the latter, the authors gives a general set of sufficient assumptions for non explosion, some of them being further generalized in [12].…”

Section: Example: Feller Process With Hard Obstaclementioning

confidence: 99%

A central limit theorem for Fleming–Viot particle systems

Cérou¹,

Delyon²,

Guyader³

et al. 2020

Ann. Inst. H. Poincaré Probab. Statist.

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“…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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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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“…This Fleming-Viot type system has been introduced by Burdzy, Holyst, Ingermann and March in [5] and studied in [6], [13], [21], [14] for multi-dimensional diffusion processes. The study of this system when the underlying Markov process X is a continuous time Markov chain in a countable state space has been initiated in [12] and followed by [1], [2], [16], [3] and [10].…”

Section: Introductionmentioning

confidence: 99%