KPP equation and supercritical branching brownian motion in the subcritical speed area. Application to spatial trees

Chauvin, Brigitte; Rouault, Alain

doi:10.1007/bf00356108

Cited by 130 publications

(163 citation statements)

References 21 publications

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“…Chauvin and Rouault [9] proved analogous results to Theorems 6-8 in the context of standard BBM. Although guided by their approach when we prove Theorem 6, there are a number of complications caused by the killing at the origin.…”

Section: Introduction and Summary Of Resultssupporting

confidence: 56%

“…The FKPP equation has been much studied by both analytic techniques, as in the original papers of Fisher [14] and Kolmogorov et al [25], as well as probabilistic methods as found in McKean [31,32], Bramson [5,6], Uchiyama [36], Neveu [33], Chauvin and Rouault [9,10], Harris [18] and Kyprianou [27], to name just a few. In addition we refer the reader to Ikeda et al [20,21,22] and Freidlin [15] for extensive discussion of the more general theory of the probabilistic representation of solutions of ordinary and partial differential equations.…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

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Further probabilistic analysis of the Fisher–Kolmogorov–Petrovskii–Piscounov equation: one sided travelling-waves

Harris

Kyprianou

2006

Annales de l'Institut Henri Poincare (B) Probability and Statis

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At the heart of this article will be the study of a branching Brownian motion (BBM) with killing, where individual particles move as Brownian motions with drift −ρ, perform dyadic branching at rate β and are killed on hitting the origin.Firstly, by considering properties of the right-most particle and the extinction probability, we will provide a probabilistic proof of the classical result that the 'one-sided' FKPP travelling-wave equation of speed −ρ with solutions f : [0, ∞) → [0, 1] satisfying f (0) = 1 and f (∞) = 0 has a unique solution with a particular asymptotic when ρ < √ 2β, and no solutions otherwise. Our analysis is in the spirit of the standard BBM studies of Harris [18] and Kyprianou [27] and includes an intuitive application of a change of measure inducing a spine decomposition that, as a by product, gives the new result that the asymptotic speed of the right-most particle in the killed BBM is √ 2β − ρ on the survival set. Secondly, we introduce and discuss the convergence of an additive martingale for the killed BBM, W λ , that appears of fundamental importance as well as facilitating some new results on the almost-sure exponential growth rate of the number of particles of speed λ ∈ (0, √ 2β − ρ). Finally, we prove a new result for the asymptotic behaviour of the probability of finding the right-most particle with speed λ > √ 2β − ρ. This result combined with Chauvin and Rouault's [9] arguments for standard BBM readily yields an analogous Yaglom-type conditional limit theorem for the killed BBM and reveals W λ as the limiting Radon-Nikodým derivative when conditioning the right-most particle to travel at speed λ into the distant future.

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Section: Introduction and Summary Of Resultssupporting

confidence: 56%

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

Further probabilistic analysis of the Fisher–Kolmogorov–Petrovskii–Piscounov equation: one sided travelling-waves

Harris

Kyprianou

2006

Annales de l'Institut Henri Poincare (B) Probability and Statis

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“…Such an interpretation of the measureQ was first given by Chauvin and Rouault [9] in the context of BBM, allowing them to come to the important conclusion that under the new measure Q the branching diffusion remains largely unaffected, except that the Brownian particles of a single (random) line of descent in the family tree are given a changed motion, with an accelerated birth rate -although they did not have random family sizes, so the size-biasing aspect was not seen. Size-biasing has been known for a long time in the study of branching populations, and in the context of spines, it was introduced in the Lyons et al papers [43,41,44].…”

Section: Understanding the Measureqmentioning

confidence: 94%

“…The work of Lyons et al [43,41,44], that of Chauvin and Rouault [9] and more recently of Kyprianou [40] suggests that when a change of measure is carried out with a branching-diffusion additive martingale like Z(t) it is typical to expect three changes: the spine will gain a drift, its fission times will be increased and the distribution of its family sizes will be size-biased. In section 6.1 we shall confirm this, but we first take a separate look at the martingales that could perform these changes, and which we shall combine to obtain a martingaleζ(t) that will ultimately be used to change the measureP .…”

Section: Definition 52mentioning

confidence: 99%

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A Spine Approach to Branching Diffusions with Applications to L p -Convergence of Martingales

Hardy

Harris

2009

Lecture Notes in Mathematics

153

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Summary. We present a modified formalization of the 'spine' change of measure approach for branching diffusions in the spirit of those found in Kyprianou [40] and Lyons et al. [44,43,41]. We use our formulation to interpret certain 'GibbsBoltzmann' weightings of particles and use this to give an intuitive proof of a general 'Many-to-One' result which enables expectations of sums over particles in the branching diffusion to be calculated purely in terms of an expectation of one 'spine' particle. We also exemplify spine proofs of the L p -convergence (p ≥ 1) of some key 'additive' martingales for three distinct models of branching diffusions, including new results for a multi-type branching Brownian motion and discussion of left-most particle speeds.

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From 1 to 6: A Finer Analysis of Perturbed Branching Brownian Motion

Bovier¹,

Hartung

2020

Comm Pure Appl Math

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We derive a multiscale generalisation of the Bakry-Émery criterion for a measure to satisfy a log-Sobolev inequality. Our criterion relies on the control of an associated PDE well-known in renormalisation theory: the Polchinski equation. It implies the usual Bakry-Émery criterion, but we show that it remains effective for measures that are far from log-concave. Indeed, using our criterion, we prove that the massive continuum sine-Gordon model with < 6 satisfies asymptotically optimal log-Sobolev inequalities for Glauber and Kawasaki dynamics. These dynamics can be seen as singular SPDEs recently constructed via regularity structures, but our results are independent of this theory.

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KPP equation and supercritical branching brownian motion in the subcritical speed area. Application to spatial trees

Cited by 130 publications

References 21 publications

Further probabilistic analysis of the Fisher–Kolmogorov–Petrovskii–Piscounov equation: one sided travelling-waves

Further probabilistic analysis of the Fisher–Kolmogorov–Petrovskii–Piscounov equation: one sided travelling-waves

A Spine Approach to Branching Diffusions with Applications to L p -Convergence of Martingales

From 1 to 6: A Finer Analysis of Perturbed Branching Brownian Motion

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