Simon C. Harris

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

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 ‘Gibbs-Boltzmann’ 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.

The many-to-few lemma and multiple spines

We develop a simple and intuitive identity for calculating expectations of weighted k-fold sums over particles in branching processes, generalising the well-known many-to-one lemma.

Algebra analysis and probability for a coupled system of reaction-diffusion 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. After § 1 a, it concentrates on the simplest case of a coupled reaction-diffusion equation. It provides a complete treatment of the existence, uniqueness, and asymptotic behaviour of monotone travelling waves to various equilibria, both by differential-equation theory and by probability theory. Each approach raises interesting questions about the other. The differential-equation treatment makes new use of the maximum principle for this type of problem. It suggests a numerical method of solution which yields computer pictures which illustrate the situation very clearly. The probabilistic treatment is careful in its proofs of martingale (as opposed to merely local-martingale) properties. A new change-of-measure technique is used to obtain the best lower bound on the speed of the monotone travelling wave with Heaviside initial conditions. Waves to different equilibria are shown to be related by Doob h-transforms. Large-deviation theory provides heuristic links between alternative descriptions of minimum wave speeds, rigorous algebraic proofs of which are provided. Since the paper was submitted, an alternative method of proving existence of monotone travelling waves has been developed by Karpelevich et al. (1993). We have extended our results in different directions from theirs (one of which is hinted at in § 1 a), and have found the methods used here well equipped for these generalizations. See the Addendum.

Branching Brownian motion with an inhomogeneous breeding potential

This article concerns branching Brownian motion (BBM) with dyadic branching at rate beta vertical bar y vertical bar(p) for a particle with spatial position y is an element of R, where beta > 0. It is known that for p > 2 the number of particles blows up almost surely in finite time, while for p = 2 the expected number of particles alive blows up in finite time, although the number of particles alive remains finite almost surely, for all time. We define the right-most particle, R-t, to be the supremum of the spatial positions of the particles alive at time t and study the asymptotics of R-t as t -> infinity. In the case of constant breeding at rate beta the linear asymptotic for R-t is long established. Here, we find asymptotic results for R-t in the case p is an element of (0, 2]. In contrast to the linear asymptotic in standard BBM we find polynomial asymptotics of arbitrarily high order as p up arrow 2, and a non-trivial limit for In R-t when p = 2. Our proofs rest on the analysis of certain additive martingales, and related spine changes of measure.

Travelling waves and homogeneous fragmentation

We formulate the notion of the classical Fisher–Kolmogorov–Petrovskii–Piscounov (FKPP) reaction diffusion equation associated with a homogeneous conservative fragmentation process and study its traveling waves. Specifically, we establish existence, uniqueness and asymptotics. In the spirit of classical works such as McKean [Comm. Pure Appl. Math. 28 (1975) 323–331] and [Comm. Pure Appl. Math. 29 (1976) 553–554], Neveu [In Seminar on Stochastic Processes (1988) 223–242 Birkhauser] and Chauvin [Ann. Probab. 19 (1991) 1195–1205], our analysis exposes the relation between traveling waves and certain additive and multiplicative martingales via laws of large numbers which have been previously studied in the context of Crump–Mode–Jagers (CMJ) processes by Nerman [Z. Wahrsch. Verw. Gebiete 57 (1981) 365–395] and in the context of fragmentation processes by Bertoin and Martinez [Adv. in Appl. Probab. 37 (2005) 553–570] and Harris, Knobloch and Kyprianou [Ann. Inst. H. Poincare Probab. Statist. 46 (2010) 119–134]. The conclusions and methodology presented here appeal to a number of concepts coming from the theory of branching random walks and branching Brownian motion (cf. Harris [Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 503–517] and Biggins and Kyprianou [Electr. J. Probab. 10 (2005) 609–631]) showing their mathematical robustness even within the context of fragmentation theory.

Branching Brownian motion in a strip: Survival near criticality

We consider a branching Brownian motion with linear drift in which particles are killed on exiting the interval (0,K) and study the evolution of the process on the event of survival as the width of the interval shrinks to the critical value at which survival is no longer possible. We combine spine techniques and a backbone decomposition to obtain exact asymptotics for the near-critical survival probability. This allows us to deduce the existence of a quasi-stationary limit result for the process conditioned on survival which reveals that the backbone thins down to a spine as we approach criticality. This paper is motivated by recent work on survival of near critical branching Brownian motion with absorption at the origin by Aidekon and Harris as well as the work of Berestycki, Berestycki and Schweinsberg.

The unscaled paths of branching Brownian motion

For a set A ⊂ C[0, ∞), we give new results on the growth of the number of particles in a 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. We also obtain new results on the number of particles near the frontier of the model. The methods used are entirely probabilistic.

Branching Brownian Motion: Almost Sure Growth Along Scaled Paths

We give a proof of a result on the growth of the number of particles along chosen paths in a branching Brownian motion. The work follows the approach of classical large deviations results, in which paths in

Scaling random walks on arbitrary sets

C[0,1]

Limiting distribution of the rightmost particle in catalytic branching Brownian motion

are rescaled onto