Measure-valued Markov processes

Dawson, Donald A.

doi:10.1007/bfb0084190

Cited by 468 publications

(333 citation statements)

References 216 publications

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“…Let q k and σ k be defined by (2.5) and (2.16) in terms of (γ k , p (k) , θ k ). We assume that {X (k) t : t ≥ 0} is a immigration particle system which satisfies 6) where (…”

Section: Stochastic Equation Of the Sdsmimentioning

confidence: 99%

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Conditional Log-Laplace Functionals of Immigration Superprocesses with Dependent Spatial Motion

Wang

Xiong

2005

Acta Appl Math

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Abstract. A non-critical branching immigration superprocess with dependent spatial motion is constructed and characterized as the solution of a stochastic equation driven by a time-space white noise and an orthogonal martingale measure. A representation of its conditional logLaplace functionals is established, which gives the uniqueness of the solution and hence its Markov property. Some properties of the superprocess including an ergodic theorem are also obtained. (2000): Primary 60J80, 60G57; Secondary 60J35 Mathematics Subject Classification

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“…Let q k and σ k be defined by (2.5) and (2.16) in terms of (γ k , p (k) , θ k ). We assume that {X (k) t : t ≥ 0} is a immigration particle system which satisfies 6) where (…”

Section: Stochastic Equation Of the Sdsmimentioning

confidence: 99%

“…Then an SDSM {X t : t ≥ 0} is characterized by the following martingale problem: For each φ ∈ C 2 b (R), Clearly, the SDSM reduces to a usual critical branching Dawson-Watanabe superprocess if h(·) ≡ 0; see e.g. Dawson [6]. A general SDSM arises as the weak limit of critical branching particle systems with dependent spatial motion.…”

Section: Introductionmentioning

confidence: 99%

Conditional Log-Laplace Functionals of Immigration Superprocesses with Dependent Spatial Motion

Wang

Xiong

2005

Acta Appl Math

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“…This process can be represented as the solution of a "formal" SPDE in the space of measures [6]. Dawson's work initiated the new research area of superprocesses (see, for example, [7]). In dimension d = 1 the formal SPDE for the Dawson-Watanabe process becomes a solvable SPDE for the density of the measure process.…”

Section: Comments On Previous Workmentioning

confidence: 99%

Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type

Kotelenez

Kurtz

2008

Probab. Theory Relat. Fields

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A class of quasilinear stochastic partial differential equations (SPDEs), driven by spatially correlated Brownian noise, is shown to become macroscopic (i.e., deterministic), as the length of the correlations tends to 0. The limit is the solution of a quasilinear partial differential equation. The quasilinear SPDEs are obtained as a continuum limit from the empirical distribution of a large number of stochastic ordinary differential equations (SODEs), coupled though a mean-field interaction and driven by correlated Brownian noise. The limit theorems are obtained by application of a general result on the convergence of exchangeable systems of processes. We also compare our approach to SODEs with the one introduced by Kunita.

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“…Then we define 12) assuming the limit exists. It should be possible to use the lace expansion to prove that the limit in (5.8) is well-defined, but this has not yet been done.…”

Section: Incipient Infinite Lattice Trees Above 8 Dimensionsmentioning

confidence: 99%

Infinite Canonical Super-Brownian Motion and Scaling Limits

Hofstad

2006

Commun. Math. Phys.

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We construct a measure valued Markov process which we call infinite canonical superBrownian motion, and which corresponds to the canonical measure of super-Brownian motion conditioned on non-extinction. Infinite canonical super-Brownian motion is a natural candidate for the scaling limit of various random branching objects on Z d when these objects are (a) critical; (b) mean-field and (c) infinite. We prove that ICSBM is the scaling limit of the spread-out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension. We conjecture that it also arises as the scaling limit in various other models above the upper-critical dimension, such as the incipient infinite lattice tree above 8 dimensions, the incipient infinite cluster for unoriented percolation, uniform spanning trees above 4 dimensions, and invasion percolation above 6 dimensions. This paper also serves as a survey of recent results linking super-Brownian to scaling limits in statistical mechanics.

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Measure-valued Markov processes

Cited by 468 publications

References 216 publications

Conditional Log-Laplace Functionals of Immigration Superprocesses with Dependent Spatial Motion

Conditional Log-Laplace Functionals of Immigration Superprocesses with Dependent Spatial Motion

Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type

Infinite Canonical Super-Brownian Motion and Scaling Limits

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