Singular Stochastic PDEs

In recent years, following the pioneering work of Fields medallist Martin Hairer, a new field of mathematics emerged at the intersection of analysis, probability theory, and mathematical physics. These developments provide the tools necessary to gain deep understanding of certain differential equations that have been derived in the physics literature to describe many interesting models such as interface growth, ferromagnetic phase transition, and stochastic fluid dynamics. However, prior to the breakthroughs of Hairer, these equations lacked a mathematical understanding, and turned out to have to be interpreted via a procedure that involves careful treatment of infinite quantities - a process that is also called renormalisation. Our proposed project aims to further develop these ideas and apply them in novel settings. For instance, what happens if the dynamics described by these differential equations are restricted to a cube or some other shape? Our recent work shows that new infinities - and thus new renormalisation - may appear on the borders of the shape for certain equations. Many other equations and/or other shapes are beyond the scope of the current theory and the general mathematical tools to treat them are waiting to be developed. In addition, what happens if the environment containing the particles of the model can change? For example, if it is influenced by the density of the particles themselves? In the particular case of a porous medium this question (establishing a solution theory for the equation of such a model) is completely open. Recently, however, a closely related area nondegenerate quasilinear equations - has seen significant developments. Using these advances, we will tackle the problem of implementing the renormalisation methods to porous medium equations and other quasilinear equations describing nonhomogeneous environments.

In recent years, following groundbreaking ideas of Fields medalist Martin Hairer, a new class of mathematical equations were brought into the spotlight of research. These so-called singular stochastic partial differential equations fall at the intersection of analysis, probability theory, and mathematical physics. They have long been know to arise in the physics literature to describe many interesting models such as interface growth, ferromagnetic phase transition, and stochastic fluid dynamics. However, prior to the breakthroughs of Hairer, these equations lacked a mathematical understanding, and turned out to have to be interpreted via a procedure that involves careful treatment of infinite quantities - a process that is also called renormalisation. This procedure is aimed to cancel out the wildest resonances in the system, which in turn are caused by the noise inherent in these equations. My project was devoted to develop these ideas and apply them in novel settings, with the goal of solving a larger class of such singular equations. One of the main directions of my research concerned what is referred to quasilinear equations. The most important novel feature of such equations is that the diffusivity of the system is no longer homogeneous. Instead, it can be thought to depend on an environment - which, in turn, is linked back to the system (that is, the solution of the equation) itself. I was interested in addressing two of the major difficulties that this type of dynamics give rise to. First, the link between the environment and the solution can be a new source of resonance - therefore, this sets up the need for a new class of renormalisation. Moreover, the diffusive part of the system no longer dominates the singular part, which is a major challenge in their analysis. In my work I developed probabilistic and combinatoric tools that can be used to handle the renormalisation of quasilinear singular equations. Second, in situations such as the well-known porous media equations, the diffusive part completely shuts down on certain regimes of the system. With my collaborators we showed that despite the lack of diffusivity, we can construct solutions to such equations as long as the noise is compatible with the almost century-long stochastic calculus of Ito.