Nonlinear partial differential equations appear naturally in many biological or chemical systems. E.g., activator-inhibitor systems play a role in morphogenesis and may generate different patterns. Noisy random fluctuations are ubiquitous in the real world. The randomness leads to various new phenomena and may have a non–trivial impact on the behaviour of the solution. The presence of the stochastic term (or noise) in the model often leads to qualitatively new types of behaviour, which helps to understand the real processes and is also often more realistic. Due to the interplay of noise and nonlinearity, noise-induced transitions, stochastic resonance, metastability, or noise-induced chaos may appear. Noise in stochastic Turing patterns expands the range of parameters in which Turing patterns appears.
The topic of the talk is a nonlinear partial differential equation disturbed by stochastic noise. Here, we will present recent results about the existence of martingale solutions using a stochastic version of a Tychanoff-Schauder type Theorem. In particular, we will introduce the stochastic Klausmeier system, a system which is not monotone, nor do they satisfy a maximum principle. So, the existence of a solution can only be shown using compactness arguments.
In the talk, we first introduce the System. Secondly, we will introduce the notion of martingale solutions and present our main result. Finally, we will outline the proof of our main result, i.e., the proof of the existence of martingale solutions.