Stochastic systems of cross-diffusion

Imagine one has a substance, let`s say a population of cells and a type of chemical, where both are spreading and interacting in a space, like a petri dish in a lab. These substances don`t just move randomly; their movement is influenced by each other. In other words, the presence of one chemical affects how the other chemical spreads and vice versa. First, how these substances spread and interact over time and space is described mathematically by a so-called "partial differential equation" . The "partial" part means that the equation involves multiple variables (like both time and space in this case), and "differential" refers to the fact that the rates of changes in space and time are involved. What does "cross-diffusion" mean? In a regular diffusion process, a substance spreads from areas of high concentration to areas of low concentration. Cross diffusion, on the other hand, involves the interaction between two or more substances as they spread. It means that the movement of one substance is influenced not only by its own concentration but also by the concentration of other substances present in the same space. In our examples, it could be that the cells are attracted by the chemical and, at the same time, produce this chemical. A situation which comes up in slime mould. So, a mathematical partial differential equation with cross-diffusion describes how multiple substances interact and spread in a given space, considering their influence on each other`s movement. Scientists use these equations to model various real-world situations, such as the diffusion of chemicals, heat, or other quantities in biological, chemical, and physical systems where interactions between different substances are essential to understanding the overall behaviour. Modelling the microscopic behaviour of irregular individual cell`s motions and the environment naturally gives rise to some intrinsic noise. From the macroscopic perspective, one interprets the movement of the cells as a result of microscopic irregular movement of a single member of the considered population of cells. One is neglecting the random fluctuations by taking the limit and passing to the macroscopic equations. In the project, we want to model systems of cross diffusions with a stochastic term, modelling the fluctuation of the cells, the molecules, and the unpredictable environment. Here, several topics will be considered, running from theoretical questions like investigating the existence of such a solution satisfying the equations, phenomena in the long term, and relation to particle systems up to more practical questions like the modelling of these systems on a computer.