Week 07.07.2025 – 13.07.2025

Nonlinear partial differential equations naturally arise in many biological and chemical systems. For example, activator–inhibitor systems play a key role in morphogenesis and can generate diverse spatial patterns. Noisy random fluctuations are ubiquitous in real-world environments. The inclusion of randomness often leads to qualitatively new phenomena and may significantly influence the behavior of solutions. The stochastic terms (i.e., noise) in such models frequently give rise to richer dynamics, offering more realistic descriptions of complex systems and aiding our understanding of the underlying processes.



The interaction between noise and nonlinearity can lead to phenomena such as noise-induced transitions, stochastic resonance, metastability, or even noise-induced chaos. In particular, the presence of noise in stochastic Turing systems can expand the range of parameters for which pattern formation occurs.



However, standard methods often fail in this context. These systems are frequently non-monotone and may not satisfy any maximum principle. Overcoming these obstacles typically requires alternative approaches. One such method relies on compactness arguments combined with a fixed point theorem.

The focus of this talk is a nonlinear partial differential equation perturbed by stochastic noise, and the methodology for proving the existence of martingale solutions via a stochastic version of a Tychonoff–Schauder-type theorem. We will begin by presenting the stochastic Schauder–Tychonoff theorem and illustrating its application through several examples. We will also outline the proof of our main result—namely, the existence theorem based on this stochastic compactness framework.