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.