Branching Markov processes I

Ikeda, Nobuyuki; Nagasawa, Masao; Watanabe, Satoshi

doi:10.1215/kjm/1250524137

Cited by 122 publications

(70 citation statements)

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“…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. Much attention has been given to FKPP solutions of the form u(t, x) = f (x − ρt) for f ∈ C 2 (R), leading to the so called FKPP travelling-wave (TW) equation…”

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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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 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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“…Before proceeding any further,following Ikeda et al [2], [3], [4], [5], we need to introduce some background related to branching Markov processes. Nagasawa and Sirao [8] …”

Section: Preliminariesmentioning

confidence: 99%

“…In fact, the authors of the first such work, Nagasawa and Sirao [8], mentioned in their introduction that they were motivated by Fujita's work. Ikeda et al [2], [3], [4], [5] gave a full description of branching Markov processes, which actually worked as a foundation for Nagasawa and Sirao to come up with the first probabilistic study of existence of global solutions and finite-time blow up for the equation…”

Section: Introductionmentioning

confidence: 99%

“…It is interesting to see if, in our case, similar conditions guarantee the existence of global positive solutions, namely, large mobility of individuals and quick decay of the initial value ensure boundedness of u(t, x) for all t ≥ 0. In the next section we introduce the ingredients we need to verify this, following Ikeda et al [2], [3], [4], [5]. In Section 3 we describe Nagasawa-Sirao's condition for the existence of global solutions, and then we consider the bounded setup, i.e.…”

Section: Introductionmentioning

confidence: 99%

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Nonexplosion of a class of semilinear equations via branching particle representations

Chakraborty¹,

López-Mimbela²

2008

Adv. Appl. Probab.

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We consider a branching particle system where an individual particle gives birth to a random number of offspring at the place where it dies. The probability distribution of the number of offspring is given by p k , k = 2, 3, . . . . The corresponding branching process is related to the semilinear partial differential equationwhere A is the infinitesimal generator of a multiplicative semigroup and the p k s, k = 2, 3, . . . , are nonnegative functions such that k p k = 1. We obtain sufficient conditions for the existence of global positive solutions to semilinear equations of this form. Our results extend previous work by Nagasawa and Sirao (1969) and others.

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“…and Watanabe [7], and later McKean [11]) use branching coupled with a diffusion, and the stochastic representation is derived directly without Fourier series, so that the linear operator A is limited to generators of diffusions. Our first aim is to show that this method applies to a large range of equations.…”

Section: Introductionmentioning

confidence: 99%

A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees

Blömker

Romito

Tribe

2007

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

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ABSTRACT. The solutions to a large class of semi-linear parabolic PDEs are given in terms of expectations of suitable functionals of a tree of branching particles. A sufficient, and in some cases necessary, condition is given for the integrability of the stochastic representation, using a companion scalar PDE.In cases where the representation fails to be integrable a sequence of pruned trees is constructed, producing a approximate stochastic representations that in some cases converge, globally in time, to the solution of the original PDE.

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Branching Markov processes I

Cited by 122 publications

References 0 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

Nonexplosion of a class of semilinear equations via branching particle representations

A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees

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