Random Travelling Waves for the KPP Equation with Noise

Mueller, Carl; Sowers, Richard B.

doi:10.1006/jfan.1995.1038

Cited by 76 publications

(95 citation statements)

References 5 publications

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“…It has been rigorously demonstrated [46] that the front generated by the sFKPP equation (7.10) is compact : For any t, there exists a x r (t) such that u(x, t) = 0 for x > x r (t), and also a x l (t) such that u(x, t) = 1 for x < x l (t). One expects a similar property for Eq.…”

Section: Some Results From Sfkpp Equation and Their Consequences For Qcdmentioning

confidence: 99%

A Langevin equation for high-energy evolution with pomeron loops

2005

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We show that the Balitsky-JIMWLK equations proposed to describe non-linear evolution in QCD at high energy fail to include the effects of fluctuations in the gluon number, and thus to correctly describe both the low density regime and the approach towards saturation. On the other hand, these fluctuations are correctly encoded (in the limit where the number of colors is large) in Mueller's color dipole picture, which however neglects saturation. By combining the dipole picture at low density with the JIMWLK evolution at high density, we construct a generalization of the Balitsky hierarchy which includes the particle number fluctuations, and thus the pomeron loops. After an additional coarse-graining in impact parameter space, this hierarchy is shown to reduce to a Langevin equation in the universality class of the stochastic Fisher-Kolmogorov-Petrovsky-Piscounov (sFKPP) equation. This equation implies that the non-linear effects in the evolution become important already in the high momentum regime where the average density is small, which signals the breakdown of the BFKL approximation.

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Section: Some Results From Sfkpp Equation and Their Consequences For Qcdmentioning

confidence: 99%

A Langevin equation for high-energy evolution with pomeron loops

2005

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“…This result does not rely on small noise but, as in [MS95], relies on the mass creation term λu − u 2 that drives the solution through space. We believe that this driving force makes the finite width of the interface more plausible.…”

Section: Introductionmentioning

confidence: 79%

“…The tools used in [MS95] are very different, and would apply to a class of equations with coefficients satisfying the same general properties. However, the result relies on taking ε small, so that the equation closely follows the underlying deterministic KPP equation over finite time intervals.…”

Section: Introductionmentioning

confidence: 99%

Finite Width For a Random Stationary Interface

Mueller

Tribe

1997

Electron. J. Probab.

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We study the asymptotic shape of the solution u(t, x) ∈ [0, 1] to a one-dimensional heat equation with a multiplicative white noise term. At time zero the solution is an interface, that is u(0, x) is 0 for all large positive x and u(0, x) is 1 for all large negative x. The special form of the noise term preserves this property at all times t ≥ 0. The main result is that, in contrast to the deterministic heat equation, the width of the interface remains stochastically bounded.

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“…This SPDE has already been investigated in Mueller and Sowers [12] in detail, where the existence of travelling waves was shown for θ 0 big enough.…”

Section: Literature Reviewmentioning

confidence: 90%

Convergence of Rescaled Competing Species Processes to a Class of SPDEs

Kliem¹

2011

Electron. J. Probab.

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One can construct a sequence of rescaled perturbations of voter processes in dimension d = 1 whose approximate densities are tight. By combining both long-range models and fixed kernel models in the perturbations and considering the critical long-range case, results of Cox and Perkins (2005) are refined. As a special case we are able to consider rescaled Lotka-Volterra models with long-range dispersal and short-range competition. In the case of long-range interactions only, the approximate densities converge to continuous space time densities which solve a class of SPDEs (stochastic partial differential equations), namely the heat equation with a class of drifts, driven by Fisher-Wright noise. If the initial condition of the limiting SPDE is integrable, weak uniqueness of the limits follows. The results obtained extend the results of Mueller and Tribe (1995) for the voter model by including perturbations. In particular, spatial versions of the Lotka-Volterra model as introduced in Neuhauser and Pacala (1999) are covered for parameters approaching one. Their model incorporates a fecundity parameter and models both intra-and interspecific competition.

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Random Travelling Waves for the KPP Equation with Noise

Cited by 76 publications

References 5 publications

A Langevin equation for high-energy evolution with pomeron loops

A Langevin equation for high-energy evolution with pomeron loops

Finite Width For a Random Stationary Interface

Convergence of Rescaled Competing Species Processes to a Class of SPDEs

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