Orateur
Description
I will present an ongoing joint research project with Harprit Singh (postdoctoral researcher at the University of Vienna) on the stochastic homogenization of parabolic singularstochastic PDEs (singular SPDEs) in correlated media. Our goal is to understand the large-scale behavior of such equations, that is, to derive an effective equation, for parabolic singularSPDEs with oscillatory random coefficients. The main challenge lies in the fact that these equations are ill-posed and require a renormalization procedure to construct a pathwise solution theory.
I will focus on the example of the $\varphi^4_2$-equation, a toy model describing the dynamics of thermodynamic fluctuations in two-dimensional ferromagnets near the critical temperature. First, I will present recent results in this direction for periodic coefficients, due to Singh and Hairer, which illustrate how homogenization and renormalization can be combined to obtain an effective, well-posed stochastic PDE. Second, I will discuss the challenges involved in extending these results to the random case and present recent progress on the renormalization of the $\varphi^4_2$-equation in a correlated random environment obtained together with Singh.