We study the asymptotic behaviour, in the small noise limit, of stochastic travelling wave solutions to reaction-diffusion equations perturbed by Wright-Fisher noise. Such equations are predicted to display three distinct responses to noise in three parametric regimes: fully pushed, semi-pushed, and pulled. We prove, for the entire fully pushed regime, that solutions are asymptotically close to a stochastic shift of the deterministic travelling wave, and characterize the limiting shift process as a Brownian motion with drift. This gives the first full fluctuation theorem demonstrating fully pushed phenomenology for a non-linear stochastic reaction-diffusion equation and verifies a physical conjecture of Birzu, Hallatschek and Korolev [BHK18]. The proof uses an infinite-dimensional version of a method introduced by Katzenberger [Kat91], as pioneered by Funaki [Fun95]. This approach views the dynamics as a stochastic perturbation of a dynamical system (the PDE) with strong drift towards an invariant manifold, in our case the set of shifts of the travelling wave profile, and gives an expression for the stochastic motion "along" this manifold. Implementing this method in our setting requires many ingredients, including a close analysis of the dynamics of the corresponding PDE, integrability and regularity properties of solutions to the SPDE, and sharp control of the position of the right endpoint of the solution's support.
We explore the impact of different forms of stochasticity on the expansion dynamics of a stochastic growth model called the ∞-parent spatial Λ-Fleming Viot process. This process belongs to a family of population genetics processes in a spatial continuum, and was recently introduced to study the evolution of genetic diversity in spatially expanding populations. Its stochastic reproduction dynamics gives rise to a rich growth structure, on which first theoretical results were obtained. In this paper, we further explore this growth dynamics using two complementary approaches: an analytical study of a simplified model for growth at the front edge, and a simulation-based study. We show that the observed expansion speed is the result of the interplay of stochasticity in shapes, timings and locations of reproduction events, each form of stochasticity being necessary but not sufficient to explain the expansion dynamics. We also identify distinctive scaling regimes for the variance of hitting times by the front and the bulk of the expansion. Moreover, we obtain results on the scaling of the front fluctuations, which point towards the front interface belonging to the KPZ universality class.
We respond to the discussion comments of the Proposer and Seconder of the vote of thanks, and to the eight other contributions discussing our work.
