Ecole d'Eté de Probabilités de Saint-Flour XXI - 1991

Dawson, Donald A.; Maisonneuve, Bernard; Spencer, Joel

doi:10.1007/bfb0084189

Cited by 196 publications

(300 citation statements)

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“…The notion of weak convergence is defined with respect to the finite measures N ε 0 (·) ≡ N 0 (·, S > ε) (see, e.g., [37]) and, in particular, nθ nt → γ −1 N 0 (S > t) = 2/(γt). See [12], [48] for detailed surveys of super-processes and convergence towards them, and see [16], [17], [44] for introductions to super-processes and continuous-state branching processes.…”

Section: Introduction and Resultsmentioning

confidence: 99%

The survival probability and r-point functions in high dimensions

Hofstad

Holmes

2013

Ann. Math.

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In this paper we investigate the survival probability, θn, in high-dimensional statistical physical models, where θn denotes the probability that the model survives up to time n. We prove that if the r-point functions scale to those of the canonical measure of super-Brownian motion, and if certain self-repellence and total-population tail-bound conditions are satisfied, then nθn → 2/(AV ), where A is the asymptotic expected number of particles alive at time n, and V is the vertex factor of the model. Our results apply to spread-out lattice trees above 8 dimensions, spread-out oriented percolation above 4 + 1 dimensions, and the spread-out contact process above 4 + 1 dimensions. In the case of oriented percolation, this reproves a result by the first author, den Hollander, and Slade (which was proved using heavy lace expansion arguments), at the cost of losing explicit error estimates. We further derive several consequences of our result involving the scaling limit of the number of particles alive at time proportional to n. Our proofs are based on simple weak convergence arguments.

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Section: Introduction and Resultsmentioning

confidence: 99%

The survival probability and r-point functions in high dimensions

Hofstad

Holmes

2013

Ann. Math.

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“…In analogy to the definition of solution spaces in the continuous setting, (11) and (12), we consider solutions with continuous paths in the space…”

Section: Formulation Of the Main Resultsmentioning

confidence: 99%

“…A number of variations of (6), also for white noise and d = 1, have been linked to generalisations of the Dawson-Watanabe superprocess and other particle systems. We refer to [11,9,13,30] for an overview and further references.…”

Section: Introductionmentioning

confidence: 99%

On Convergence of Population Processes in Random Environments to the Stochastic Heat Equation with Colored Noise

Sturm

2003

Electron. J. Probab.

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We consider the stochastic heat equation with a multiplicative colored noise term on R d for d ≥ 1. First, we prove convergence of a branching particle system in a random environment to this stochastic heat equation with linear noise coefficients. For this stochastic partial differential equation with more general non-Lipschitz noise coefficients we show convergence of associated lattice systems, which are infinite dimensional stochastic differential equations with correlated noise terms, provided that uniqueness of the limit is known. In the course of the proof, we establish existence and uniqueness of solutions to the lattice systems, as well as a new existence result for solutions to the stochastic heat equation. The latter are shown to be jointly continuous in time and space under some mild additional assumptions.

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“…9) almost surely for all t > 1. We do not know if these bounds are sharp, only that lim sup 10) with positive probability in all dimensions d 1 and for all λ > 0. However, our methods are in some sense robust: We will prove that some aspects of (1.…”

Section: Introductionmentioning

confidence: 96%

Dissipation and high disorder

et al. 2017

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Given a field {B(x)} x∈Z d of independent standard Brownian motions, indexed by Z d , the generator of a suitable Markov process on Z d , G, and sufficiently nice function σ : [0, ∞) → [0, ∞), we consider the influence of the parameter λ on the behavior of the system,We show that for any λ > 0 in dimensions one and two the total mass x∈Z d u t (x) → 0 as t → ∞ while for dimensions greater than two there is a phase transition point λ c ∈ (0, ∞) such that for λ > λ c , Z d u t (x) → 0 as t → ∞ while for λ < λ c , Z d u t (x) → 0 as t → ∞.

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Ecole d'Eté de Probabilités de Saint-Flour XXI - 1991

Cited by 196 publications

References 0 publications

The survival probability and r-point functions in high dimensions

The survival probability and r-point functions in high dimensions

On Convergence of Population Processes in Random Environments to the Stochastic Heat Equation with Colored Noise

Dissipation and high disorder

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