arXiv subjects

Simon C. Harris

Publications and source records attributed to Simon C. Harris.

26 records · Page 2Linked to original sources

arXiv2014

Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential

We consider a branching particle system where each particle moves as an independent Brownian motion and breeds at a rate proportional to its distance from the origin raised to the power $p$, for $p\in[0,2)$. The asymptotic behaviour of the right-most particle for this system is already known; in this article we give large deviations probabilities for particles following "difficult" paths, growth rates along "easy" paths, the total population growth rate, and we derive the optimal paths which particles must follow to achieve this growth rate.

math.PR↗

arXiv2013

A strong law of large numbers for branching processes: almost sure spine events

We demonstrate a novel strong law of large numbers for branching processes, with a simple proof via measure-theoretic manipulations and spine theory. Roughly speaking, any sequence of events that eventually occurs almost surely for the spine entails the almost sure convergence of a certain sum over particles in the population.

math.PR↗

arXiv2013

Branching Random Walk in an inhomogeneous breeding potential

We consider a continuous-time branching random walk in the inhomogeneous breeding potential $β|.|^p$, where $β> 0$, $p \geq 0$. We prove that the population almost surely explodes in finite time if $p > 1$ and doesn't explode if $p \leq 1$. In the non-explosive cases, we determine the asymptotic behaviour of the rightmost particle.

math.PR↗

arXiv2013

Branching Brownian Motion with catalytic branching at the origin

We consider a branching Brownian motion in which binary fission takes place only when particles are at the origin at a rate β> 0 on the local time scale. We obtain results regarding the asymptotic behaviour of the number of particles above λt at time t, for λ> 0. As a corollary, we establish the almost sure asymptotic speed of the rightmost particle. We also prove a Strong Law of Large Numbers for this catalytic branching Brownian motion.

math.PR↗

arXiv2010

The unscaled paths of branching Brownian motion

For a set $A\subset C[0,\infty)$, we give new results on the growth of the number of particles in a dyadic branching Brownian motion whose paths fall within A. We show that it is possible to work without rescaling the paths. We give large deviations probabilities as well as a more sophisticated proof of a result on growth in the number of particles along certain sets of paths. Our results reveal that the number of particles can oscillate dramatically. As a byproduct of our methods we also obtain new results on the number of particles near the frontier of the model. The methods used are entirely probabilistic.

math.PR↗

arXiv2007

Strong Law of Large Numbers for branching diffusions

Let $X$ be the branching particle diffusion corresponding to the operator $Lu+β(u^{2}-u)$ on $D\subseteq \mathbb{R}^{d}$ (where $β\geq 0$ and $β\not\equiv 0$). Let $λ_{c}$ denote the generalized principal eigenvalue for the operator $L+β$ on $D$ and assume that it is finite. When $λ_{c}>0$ and $L+β-λ_{c}$ satisfies certain spectral theoretical conditions, we prove that the random measure $\exp \{-λ_{c}t\}X_{t}$ converges almost surely in the vague topology as $t$ tends to infinity. This result is motivated by a cluster of articles due to Asmussen and Hering dating from the mid-seventies as well as the more recent work concerning analogous results for superdiffusions of \cite{ET,EW}. We extend significantly the results in \cite{AH76,AH77} and include some key examples of the branching process literature. As far as the proofs are concerned, we appeal to modern techniques concerning martingales and `spine' decompositions or `immortal particle pictures'.

math.PR↗

arXiv2006

A new formulation of the spine approach to branching diffusions

We present a formalization of the spine change of measure approach for branching diffusions that improves on the scheme laid out for branching Brownian motion in Kyprianou (2004) ["Travelling wave solutions to the KPP equation, Ann. Inst. H. Poincare Probab. Statist. 40, no.1, pp53-72] which itself made use of earlier works of Lyons et al (1997) ["A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes", Classical and modern branching processes, IMA Vol. Math. Appl., vol.84, Springer, New York, pp181-185]. We use our new formulation to interpret certain `Gibbs-Boltzmann' weightings of particles and use this to give a new, intuitive and proof of a more 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 particle. Significantly, our formalization has provided the foundations that facilitate a variety of new, greatly simplified and more intuitive proofs in branching diffusions: see, for example, the L^p convergence of additive martingales in Hardy and Harris (2006) ["Spine proofs for L^p-convergence of branching-diffusion martingales", arXiv:math.PR/0611056], the path large deviation results for branching Brownian motion in Hardy and Harris (2006) ["A conceptual approach to a path result for branching Brownian motion", Stochastic Processes and their Applications, doi:10.1016/j.spa.2006.05.010] and the large deviations for a continuous-typed branching diffusion in Git et al (2006) ["Exponential growth rates in a typed branching diffusion", Annals Applied Prob., (under revision)] and Hardy and Harris (2004) ["A spine proof of a lower-bound for a typed branching diffusion", no.0408, Mathematics Preprint, University of Bath].

math.PR↗

arXiv2006

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 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. The spine techniques we develop give clear and simple arguments in the spirit of the conceptual spine proofs found in Kyprianou (2004) ["Travelling wave solutions to the KPP equation, Ann. Inst. H. Poincare Probab. Statist. 40, no.1, pp53-72] and Lyons et al (1997) ["A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes", Classical and modern branching processes, IMA Vol. Math. Appl., vol.84, Springer, New York, pp181-185], and they should also extend to more general classes of branching diffusions. Importantly, the techniques in this paper also pave the way for the path large-deviation results for branching diffusions found in Hardy and Harris (2006) ["A conceptual approach to a path result for branching Brownian motion", Stochastic Processes and their Applications, doi:10.1016/j.spa.2006.05.010].

math.PR↗