Stochastic PDEs and Renormalisation

Differential equations very often describe systems that change in time: think of the motion of planets, for example. At each point in time the state of the system can be described by a few numbers (e.g., the spatial coordinates of position or velocity). Partial differential equations (PDEs) are more complex and need infinitely many numbers to describe them: for example, to describe the changes of waves in a pond, at any time we must specify the height of the water in every position simultaneously. Stochastic PDEs go one step further and also incorporate randomness: where some internal or external uncertainty perturbs the evolution of the system. They can be applied to a wide variety of models in mathematical physics. For example, the unpredictability in complex interface growth (think of the evolution of the boundary of a growing coffee stain or a forest fire) or in the movement of polymers can be described by stochastic PDEs. In order to be able to take randomness into account in the world of differential equations, one must develop completely new mathematical tools. For example, it can happen that certain terms of these differential equations become infinitely large. Conventional mathematical methods fail in this situation. However, there are so-called renormalisation methods with which one can still determine reliable results even in this case. Our project will bring new perspectives to the mathematical foundations of stochastic PDEs. With novel mathematical techniques, we will answer fundamental questions about such equations: When do solutions exist? What do they look like? How can one simulate them? How are the infinities tamed both in theory and in computation? These results will shed new light to both the mathematics of stochastic PDEs and to the underlying physical models.