The KPP equation with noise - a model exhibiting local, density dependent competition
Final Report Abstract
In this project we obtained new insights on the dynamics of solutions to the Kolmogorov-Petrovskii- Piskunov-(KPP)-equation with noise, ∂t u = ∂xx u + θu − u^2 + u^1/2 dW, t > 0, x ∈ R, u(0, x) = u0 (x) ≥ 0, where W = W (t, x) is space-time white noise and θ > 0 a parameter. Here, the focus was on the behavior of solutions as time goes to infinity. The competition term in the stochastic partial differential equation (SPDE) introduces nontrivial dependence in space and thereby makes the model a very challenging one. Additional difficulties arise due to the moderate size of the parameter θ and that due to the stochastic part, solutions are not uniformly bounded and the only steady solution is 0. As established by C. Mueller and R. Tribe (1995), solutions arise as the (weak) limit of approximate densities of occupied sites in rescaled one-dimensional long range contact processes. When investigating solutions to the KPP equation with noise, it is only natural to anticipate behavior similar in spirit to the approximating systems. Due to the long range interaction and the lack of a dual process, results for long range contact processes are limited. More is known for the nearest-neighbor contact process on Z, which served as inspiration for our proofs relating to edge speeds. Another avenue for future work is therefore to investigate in how far our techniques can in turn lead to major insights into the behavior of long range contact processes near criticality. The strategies we developed are a novel means to investigate the dynamics of stochastic processes in continuous space settings such as SPDEs. In particular, they suggest that when generalizing results from interacting particle systems based on right edge techniques to long range or SPDE setups, the right edges should not be replaced by right markers but instead by their averages over time and / or traveling waves. As a result, we were able to answer an open question on the existence of traveling waves with positive speed. Starting in the special case of Heavyside-initial data, with the parameter θ arbitrarily fixed above the critical value θc for survival, the speed of the right front of the solution is shown to be deterministic and positive. We also established that for all parameters above the critical value, there exist stochastic wavelike solutions which travel with the same deterministic positive linear speed. The main question of this project, that is, if local survival is possible and in particular, if complete convergence holds for all θ > θc was partially answered. Based on the assumption that the right front of the solution to the KPP equation with noise travels at positive linear speed, a comparison with M -dependent percolation yields “uniform distribution” in space for t → ∞ of solutions to the KPP equation with noise. This in turn will yield complete convergence. We thus derived a sufficient condition for complete convergence of solutions to the KPP equation with noise that reveals yet another link between front speeds and convergence properties of solutions. In case θ is big enough, the assumption is fulfilled and complete convergence holds. A generalized formulation of the properties of the process which lie at the heart of the construction of dynamic comparison that we introduced, allows for applying our techniques to other processes as well. Finally, first progress has been made towards finding a particle representation of solutions to the KPP equation with noise, where rather one-to-one interactions between particles “that meet in space” are modeled than long range interactions in the spirit of C. Mueller and R. Tribe, 1995. They can be a useful device in the investigation of path-properties and the longterm behavior of the system. In a collaboration with V.C. Tran, A. Blancas, S. Gufler and A. Wakolbinger, the lookdown representations of Donnelly and Kurtz were extended to branching populations under selection and competition. In the end, for technical reasons, we nevertheless chose Poisson representations as a starting point. In a first step, we discovered a more intuitive derivation of the latter constructions with the goal to make them easier accessible to a broader audience. Branching particle systems with non-bounded local downward drift modeling competition are limited in number and a topic of independent interest.
Publications
- Right marker speeds of solutions to the KPP equation with noise, Ann. Appl. Probab., 29 (6), 3637– 3694, 2019
S. Kliem
(See online at https://doi.org/10.1214/19-AAP1489) - The Genealogy of Extremal Particles of Branching Brownian Motion, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 38, 151–207, 2020
S. Kliem and K. Saha
(See online at https://doi.org/10.1142/9789811206092_0004)
