Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

Abstract

In this review, we provide a concise summary of several important mathematical results for stochastic travelling waves generated by monostable and bistable reaction-diffusion stochastic partial differential equations (SPDEs). In particular, this survey is intended for readers new to the topic but who have some knowledge in any sub-field of differential equations. The aim is to bridge different backgrounds and to identify the most important common principles and techniques currently applied to the analysis of stochastic travelling wave problems. Monostable and bistable reaction terms are found in prototypical dissipative travelling wave problems, which have already guided the deterministic theory. Hence, we expect that these terms are also crucial in the stochastic setting to understand effects and to develop techniques. The survey also provides an outlook, suggests some open problems, and points out connections to results in physics as well as to other active research directions in SPDEs.

Travelling waves for reaction–diffusion equations forced by translation invariant noise

“…The results in [33] do use cylindrical Q-Wiener processes for waves in the Fisher-KPP equation, but there the smooth covariance function q is replaced by a delta-function in order to model noise that is white in space and time. A more detailed overview of results on stochastic travelling waves can be found in the review by Kuehn [27]. Turning to non-rigorous results for (1.1) from other fields, we refer to [14] for an interesting overview of studies that have appeared in the physics and chemistry literature.…”

The wave speed of an FKPP equation with jumps via coordinated branching

“…Now, it is generally understood that in the present case, in which the nonlinearity in (1.2) is concave, the wave speed is determined by the growth of the linearisation near u = 0 of the equation: this is referred to as the pulled regime [16]. Moreover, the growth of the linearisation is roughly equivalent to that of the dual process.…”

“…The size of the gap between the speed s of the stochastic equation and the speed √ 2r of the associated deterministic one depends on the nature of the noise. In a so-called pushed regime [16] (for example in presence of a genetic drift term) the effect can be surprisingly strong also for small noise, as demonstrated in the seminal work by Mueller, Mytnik and Quastel [20]. In our case, the nonlinearity is smooth and concave: We are in the pulled regime, where the effect of noise is weaker and most importantly the speed of the wave is governed by the linearisation of the equation near u = 0.…”

Spatial dynamics of a population in a heterogeneous environment

“…Many microscopic models of front propagation (in homogeneous environment) can be seen as noisy versions of the Fisher-KPP equation, see e.g. the reviews [37,29]. A rich theory originating in the work of Brunet, Derrida and co-authors [13,14,15] has put forward some universal asymptotic behavior when the population density K goes to infinity.…”