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

@article{Harris2006FurtherPA,
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},
year={2006},
volume={42},
pages={125-145}
}
• Published 2006
• Mathematics
• Annales De L Institut Henri Poincare-probabilites Et Statistiques
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## References

SHOWING 1-10 OF 59 REFERENCES

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

• Mathematics
Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences
• 1995
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

• Mathematics
• 1990
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

• Mathematics
• 1988
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}} =

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

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• Mathematics