A Fleming–Viot Particle Representation¶of the Dirichlet Laplacian

Burdzy, Krzysztof; Hołyst, Robert; March, Peter

doi:10.1007/s002200000294

Cited by 106 publications

(211 citation statements)

References 17 publications

Supporting

Mentioning

205

Contrasting

Order By: Relevance

“…In their model N brownian particles evolve independently until one of them reaches the boundary, which plays the role of state 0. In [1], the authors prove that the empirical profile of the invariant measure converges in this case to the first eigenfunction of the Laplacian on the domain with homogeneous Dirichlet boundary conditions.…”

Section: Introductionmentioning

confidence: 97%

“…In words, each particle moves independently of the others as a continuous time Markov process with rates Q, but when it attemps to jump to state 0, it comes back immediately to Λ by jumping to the position of one of the other particles chosen uniformly at random. This type of fv process was introduced by Burdzy, Holyst and March in [1] for Brownian motions on a bounded domain. In their model N brownian particles evolve independently until one of them reaches the boundary, which plays the role of state 0.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces

2011

View full text Add to dashboard Cite

Consider a continuous time Markov chain with rates Q in the state space Λ ∪ {0}. In the associated Fleming-Viot process N particles evolve independently in Λ with rates Q until one of them attempts to jump to state 0. At this moment the particle comes back to Λ instantaneously, by jumping to one of the positions of the other particles, chosen uniformly at random. When Λ 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 quasi-stationary distribution of the one-particle motion. A key element of the approach is to show that the two-particle correlations is of order 1/N .

show abstract

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces

2011

View full text Add to dashboard Cite

show abstract

“…The first part [11] looks at a number of models that need a finite number of jumps before entering a certain center of the state space (a small set in the sense of Doeblin theory). This paper is dedicated to the harder example of the N particle system with Fleming-Viot dynamics introduced in [3] for Brownian motions. Similarly to the Wright-Fisher model, a killed particle is replaced by having one of the surviving particles branch; this can be interpreted as a jump to the location of one of the survivors, chosen uniformly.…”

Section: Introductionmentioning

confidence: 99%

Immortal particle for a catalytic branching process

Grigorescu

Kang

2011

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

Abstract. We study the existence and some asymptotic properties of a conservative branching particle system for which birth and death are triggered by contact with a set.Sufficient conditions for the process to be non-explosive are given, solving a long standing open problem. With probability one, it is shown that only one ancestry survives. In special cases, the evolution of the surviving particle is studied and for a two particle system on a half line we derive explicitly the transition function of a chain representing the position at successive branching times.

show abstract

“…As soon as one of the particles reaches the boundary ∂D, an independent Brownian particle is created at one of the sites of the survivors, chosen with uniform probability. The model (in lattice and continuous version) is due to Burdzy, Ho lyst, Ingerman and March in [2], and later studied (in its present continuous version) by some of the same authors in [3], where a law of large numbers is established for the empirical measures at fixed times. The present paper continues the investigation from [11] where a hydrodynamic limit for the joint law of the empirical measure and the average number of redistributions (boundary hits) was proven.…”

Section: Introductionmentioning

confidence: 99%

Tagged Particle Limit for a Fleming-Viot Type System

Grigorescu

Kang

2006

Electron. J. Probab.

View full text Add to dashboard Cite

Abstract. We consider a branching system of N Brownian particles evolving independently in a domain D during any time interval between boundary hits. As soon as one particle reaches the boundary it is killed and one of the other particles splits into two independent particles, the complement of the set D acting as a catalyst or hard obstacle. We determine the exact law of the tagged particle as N approaches infinity. In addition, we show that any finite number of labelled particles become independent in the limit. Both results can be seen as scaling limits of a genome population undergoing redistribution present in the Fleming-Viot dynamics.

show abstract

A Fleming–Viot Particle Representation¶of the Dirichlet Laplacian

Cited by 106 publications

References 17 publications

Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces

Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces

Immortal particle for a catalytic branching process

Tagged Particle Limit for a Fleming-Viot Type System

Contact Info

Product

Resources

About