Finite Width For a Random Stationary Interface

Mueller, Carl; Tribe, Roger

doi:10.1214/ejp.v2-21

Cited by 8 publications

(8 citation statements)

References 16 publications

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“…In the case ̺ = −1, the authors in [22] exploit the corresponding fourth moment bound to get an estimate on the moments of the width of the interface |R(u t , v t ) − L(u t , v t )|, without any rescaling [here we use the notation (5)]. However, this estimate heavily relies on the fact that there are "no holes" in the system where both u and v are zero.…”

Section: 2mentioning

confidence: 99%

“…In order to prove that the interface shrinks to one point, a possible line of attack would be to establish stationarity of the interface without any rescaling as in [22]. Another approach, which might also shed some light on the question of an explicit equation for the limit, could be to diffusively rescale the discrete-space infinite rate model and to investigate whether it converges to our limit process.…”

Section: 2mentioning

confidence: 99%

“…Therefore, we write (u t , v t ) := (u [1] t , v [1] t ) and (R t , L t ) := (R [1] t , L [1] t ). We recall from (22) and (28) in the proof of Proposition 2.2 (for the system with branching rate 1) that since ̺ < − 1 √ 2 , for anyε ∈ (0, 1 2 ) there exists a constant C(̺,ε) > 0 such that for all z > 0 and t ≥ 0,…”

mentioning

confidence: 99%

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The scaling limit of the interface of the continuous-space symbiotic branching model

2016

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The continuous-space symbiotic branching model describes the evolution of two interacting populations that can reproduce locally only in the simultaneous presence of each other. If started with complementary Heaviside initial conditions, the interface where both populations coexist remains compact. Together with a diffusive scaling property, this suggests the presence of an interesting scaling limit. Indeed, in the present paper, we show weak convergence of the diffusively rescaled populations as measure-valued processes in the Skorokhod, respectively the Meyer-Zheng, topology (for suitable parameter ranges). The limit can be characterized as the unique solution to a martingale problem and satisfies a "separation of types" property. This provides an important step toward an understanding of the scaling limit for the interface. As a corollary, we obtain an estimate on the moments of the width of an approximate interface. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2016, Vol. 44, No. 2, 807-866. This reprint differs from the original in pagination and typographic detail. 1 4 J. BLATH, M. HAMMER AND M. ORTGIESE before Corollary 1.2. Recently, analogous results have been derived by Döring and Mytnik in the case ̺ ∈ (−1, 1) in [9, 10]. Returning to the continuous-space set-up, for ̺ = −1 (the stepping stone model) Tribe [27] proves a "functional limit theorem": For a pair of (continuous) functions (u, v), define R(u, v) := sup{x : u(x) > 0}, L(u, v) = inf{x : v(x) > 0}. (5)Note that for a solution (u t , v t ) t≥0 of the symbiotic branching model, the interface at time t is contained in the interval [L(u t , v t ), R(u t , v t )]. It is proved in [27] for ̺ = −1 and for continuous initial conditions u 0 = 1 − v 0 which satisfy −∞ < L(u 0 , v 0 ) ≤ R(u 0 , v 0 ) < ∞ that under Brownian rescaling, the motion of the position of the right endpoint of the interface t → 1 n R(u n 2 t , 1 − u n 2 t ), t ≥ 0, converges to a Brownian motion as n → ∞.The above results suggest the existence of an interesting diffusive scaling limit for the continuous-space symbiotic branching model (and its interface) for ̺ > −1. This is the starting point of our investigation. However, compared to the case ̺ = −1, the situation is more involved here: For example, the total mass of the solution is not necessarily bounded, and in particular, moments of the solution may diverge as t → ∞, depending on ̺. For instance, second moments diverge for ̺ ≥ 0. In order to state this result, which was obtained in [3]

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Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

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The scaling limit of the interface of the continuous-space symbiotic branching model

2016

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“…There is a travelling wave solution to (3) (Mueller and Sowers, 1995), for which, in contrast to the classical Fisher wave, the region in which p / ∈ {0, 1} is bounded. Indeed, even in the case without selection, if we start from an initial condition in which the 'interface' between the two types is bounded, then the region in which p(t, y) / ∈ {0, 1} remains bounded for all time (Mueller and Tribe, 1997).…”

Section: One Dimensional Waves Of Advancementioning

confidence: 99%

Genetic hitchhiking in spatially extended populations

Barton

Etheridge

Kelleher

et al. 2013

Theoretical Population Biology

100

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“…Note that our results give only limited information about the shape of the interface. For the case ̺ = −1, that is, with locally constant total population size, it is shown in [16] that there exists a unique stationary interface law, which may therefore be interpreted as a "stationary wave" whose position fluctuates at the boundaries, according to [24], like a Brownian motion, hence explaining the square-root speed (note that for both results, suitable bounds on fourth mixed moments are required). However, for ̺ > −1, the population sizes of the interface are expected to fluctuate significantly and it seems unclear how this affects the shape and speed of the interface, in particular the formation of a "stationary wave."…”

Section: Longtime Behavior Of Momentsmentioning

confidence: 99%

On the moments and the interface of the symbiotic branching model

Blath¹,

Döring²,

Etheridge³

2011

Ann. Probab.

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In this paper we introduce a critical curve separating the asymptotic behavior of the moments of the symbiotic branching model, introduced by Etheridge and Fleischmann [Stochastic Process. Appl. 114 (2004) 127--160] into two regimes. Using arguments based on two different dualities and a classical result of Spitzer [Trans. Amer. Math. Soc. 87 (1958) 187--197] on the exit-time of a planar Brownian motion from a wedge, we prove that the parameter governing the model provides regimes of bounded and exponentially growing moments separated by subexponential growth. The moments turn out to be closely linked to the limiting distribution as time tends to infinity. The limiting distribution can be derived by a self-duality argument extending a result of Dawson and Perkins [Ann. Probab. 26 (1998) 1088--1138] for the mutually catalytic branching model. As an application, we show how a bound on the 35th moment improves the result of Etheridge and Fleischmann [Stochastic Process. Appl. 114 (2004) 127--160] on the speed of the propagation of the interface of the symbiotic branching model.Comment: Published in at http://dx.doi.org/10.1214/10-AOP543 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Finite Width For a Random Stationary Interface

Cited by 8 publications

References 16 publications

The scaling limit of the interface of the continuous-space symbiotic branching model

The scaling limit of the interface of the continuous-space symbiotic branching model

Genetic hitchhiking in spatially extended populations

On the moments and the interface of the symbiotic branching model

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