Stochastic Analysis Seminar

In recent years, a more top-down approach to renormalisation for singular SPDEs has emerged within the theory of regularity structures, based on regularity structures of multi-indices. This approach adopts a geometric viewpoint, aiming to stably parametrise the solution manifold rather than the larger space of renormalised objects that typically arise in fixed-point formulations of the equation. While several works have established the construction of the renormalised data (the model) in this setting, less has been shown with regards to the corresponding solution theory since the intrinsic nature of the model leads to renormalised data that is too lean to apply Hairer’s fixed-point approach.

In this talk, I will discuss past and ongoing work with L. Broux and F. Otto addressing this issue for the Phi^4 equation in its full subcritical regime. We establish local and global well-posedness within the framework of regularity structures of multi-indices; first in a space-time periodic setting and subsequently in domains with Dirichlet boundary conditions.

TBA

I will discuss a 2-dimensional model of random walk in random environment known as line model. The environment is described by two independent families of i.i.d. random variables dictating rates of jumps in vertical, respectively horizontal directions, and whose values are constant along vertical, respect. horizontal lines. When jump rates are heavy-tailed in one of the directions, the random walk becomes superdiffusive in that direction, with an explicit scaling limit written as a two-dimensional Brownian motion time-changed (in one of the components) by a process introduced by Kesten and Spitzer in 1979. I will present ideas of the proof of this result, which relies on appropriate time-change arguments.  In the case of a fully degenerate environment, I will present a sufficient condition for non-explosion of the process (which is also believed to be sharp), as well as conjectures on the associated scaling limit.

This is based on joint work with J.-D. Deuschel (TU Berlin).