Exponential growth rates & large deviations for a typed branching diffusion Y. Git * , J. W. Harris † and S. C. Harris ‡ February 26, 2004 Abstract We find the almost sure rate of exponential growth of particles, D(γ,κ), which are found simul- taneously with spatial positions near γt and type positions near κ √ t at large times t in the high temperature phase of a typed branching diffusion initially studied in Harris and Williams (1996). Our proofs rest on the study of certain martingales associated with the branching diffusion and a change of measure inducing a spine decomposition. Contents 1 Introduction and summary 2 1.1 The branching model ....................................... 2 1.2 The asymptotic growth rate of particles along spatial rays................... 3 1.3 The asymptotic shape and growth of the branching diffusion ................. 4 2 Some expectation calculations 5 2.1 The expected rate of growth along spatial rays ........................ 6 2.2 The expected asymptotic shape ................................. 7 3 Large deviation heuristics 9 3.1 The long tread .......................................... 9 3.2 The short climb .......................................... 9 3.3 A birth-death process ...................................... 10 3.4 Finding the optimal path and probability ........................... 11 3.5 A note on the optimal paths ................................... 12 3.6 The successful deviant particles ................................. 13 4 Martingales, filtrations and changes of measure 13 5 Heuristics with spines 19 6 Proof of Theorem 14 23 7 Martingale results 28 7.1 The ‘ground-state’ martingales ................................. 29 7.2 The ‘excited-state’ martingales and an important convergence theorem ........... 29 7.3 Rates of convergence to zero ................................... 30 7.4 Martingales at the ‘critical’ parameter value .......................... 31 * Yoav Git, 22 Mill Street, Cambridge, CB1 2HP. email: Y.Git@statslab.cam.ac.uk † Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, U.K.. email: j.w.harris@bath.ac.uk ‡ Correspondence: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, U.K.. email: s.c.harris@bath.ac.uk 1