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

  title={Further probabilistic analysis of the Fisher–Kolmogorov–Petrovskii–Piscounov equation: one sided travelling-waves},
  author={J. W. Harris and Simon R. Harris and Andreas E. Kyprianou},
  journal={Annales De L Institut Henri Poincare-probabilites Et Statistiques},

Survival of Near-Critical Branching Brownian Motion

Consider a system of particles performing branching Brownian motion with negative drift $\mu= \sqrt{2 - \varepsilon}$ and killed upon hitting zero. Initially there is one particle at x>0. Kesten

Branching Brownian motion with selection

In this thesis, branching Brownian motion (BBM) is a random particle system where the particles diffuse on the real line according to Brownian motions and branch at constant rate into a random number

A Strong Law of Large Numbers for Super-Critical Branching Brownian Motion with Absorption

We consider a (one-dimensional) branching Brownian motion process with a general offspring distribution having at least two moments, and in which all particles have a drift towards the origin where

Yaglom-type limit theorems for branching Brownian motion with absorption

We consider one-dimensional branching Brownian motion in which particles are absorbed at the origin. We assume that when a particle branches, the offspring distribution is supercritical, but the

Spine proofs for Lp-convergence of branching-diffusion martingales

Using the foundations laid down in Hardy and Harris (2006) ["A new formulation of the spine approach in branching diffusions", arXiv:math.PR/0611054], we present new spine proofs of the

Speed and fluctuations of N-particle branching Brownian motion with spatial selection

We consider branching Brownian motion on the real line with the following selection mechanism: every time the number of particles exceeds a (large) given number N, only the N right-most particles are

Genealogy of extremal particles of branching Brownian motion

Branching Brownian motion describes a system of particles that diffuse in space and split into offspring according to a certain random mechanism. By virtue of the groundbreaking work by M. Bramson on

Velocity of the $L$-branching Brownian motion

We consider a branching-selection system of particles on the real line that evolves according to the following rules: each particle moves according to a Brownian motion during an exponential lifetime

Branching Brownian motion in an expanding ball and application to the mild obstacle problem

We first consider a d-dimensional branching Brownian motion (BBM) evolving in an expanding ball, where the particles are killed at the boundary of the ball, and the expansion is subdiffusive in time.

Branching Brownian motion seen from its tip

It has been conjectured since the work of Lalley and Sellke (Ann. Probab., 15, 1052–1061, 1987) that branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an



Algebra, analysis and probability for a coupled system of reaction-duffusion equations

This paper is designed to interest analysts and probabilists in the methods of the ‘other’ field applied to a problem important in biology and in other contexts. It does not strive for generality.

Travelling-waves for the FKPP equation via probabilistic arguments

  • S. R. Harris
  • Mathematics
    Proceedings of the Royal Society of Edinburgh: Section A Mathematics
  • 1999
We outline a completely probabilistic study of travelling-wave solutions of the FKPP reaction-diffusion equation that are monotone and connect 0 to 1. The necessary asymptotics of such

Supercritical Branching Brownian Motion and K-P-P Equation In the Critical Speed-Area

If Rt is the position of the rightmost particle at time t in a one dimensional branching brownian motion, whore α is the inverse of the mean life time and m is the mean of the

Maximal displacement of branching brownian motion

It is shown that the position of any fixed percentile of the maximal displacement of standard branching Brownian motion in one dimension is 21/2t–3 · 2−3/2 log t + O(1) at time t, the second-order

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

SummaryIf Rt is the position of the rightmost particle at time t in a one dimensional branching brownian motion, u(t, x)=P(Rt>x) is a solution of KPP equation: $$\frac{{\partial u}}{{\partial t}} =

Branching brownian motion with absorption

On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type

CONTENTS Page Introduction 11 Part I. Travelling waves 1. Existence, uniqueness and properties of the travelling wave 17 2. KPP transform of the travelling wave 24 3. Second existence theorem for the

Measure change in multitype branching

The Kesten-Stigum theorem for the one-type Galton-Watson process gives necessary and sufficient conditions for mean convergence of the martingale formed by the population size normed by its

Local extinction versus local exponential growth for spatial branching processes

Let X be either the branching diffusion corresponding to the operator Lu+β(u2−u) on D⊆ Rd [where β(x)≥0 and β≡0 is bounded from above] or the superprocess corresponding to the operator Lu+βu−αu2 on