Week 02.10.2023 – 08.10.2023

Cylindrical Lévy processes are a natural extension of cylindrical Brownian motion which has been the standard model of random perturbations of partial differential equations and other models in infinite dimensions for the last 50 years. Here, the attribute cylindrical refers to the fact that cylindrical Brownian motions are not classical stochastic processes attaining values in the underlying space but are generalised objects. The reasons for the choice of cylindrical but not classical Brownian motion can be found in the facts that there does not exist a classical Brownian motion with independent components in an infinite dimensional Hilbert space, and that cylindrical processes enable a very flexible modelling of random noise in time and space.

This talk is a very introductory presentation to cylindrical Lévy processes. We explain the difficulty to define random noises in infinite dimensions and explain the approach by cylindrical measures and cylindrical random variables, which are strongly related to other areas such as harmonic analysis and operator theory. We present some specific examples of cylindrical Lévy processes in detail and discuss their relations to other models of random perturbations in the literature. We explain how a theory of stochastic integration for cylindrical Lévy processes can be developed although standard approaches to stochastic integration cannot be applied, and how this theory can be used to derive a theory of stochastic partial differential equations driven by Cylindrical Lévy processes.

We will review some general concepts of deformation theory. Then we will apply these ideas in order to explore the geometry of the moduli space Inv of foliations on a given variety $X$ around the points corresponding to foliations induced by Lie group actions. More precisely, let $X$ be a smooth projective variety over the complex numbers and $S(d)$ the scheme parametrizing $d$-dimensional Lie subalgebras of $H^0(X,\mathcal{T} X)$. For every $\mathfrak{g} \in S(d)$ one can consider the corresponding element $\mathcal{F}(\mathfrak{g})\in Inv$, whose generic leaf coincides with an orbit of the action of $\exp(\mathfrak{g})$ on $X$. We will show that under mild hypotheses, after taking a stratification $\coprod_i S(d)_i\to S(d)$ this assignment yields an isomorphism $\coprod_i S(d)_i\to Inv$ locally around $\mathfrak{g}$ and $\mathcal{F}(\mathfrak{g})$.

The non-minimal coupling of scalar fields to gravity has been claimed to violate energy conditions,
leading to exotic phenomena such as traversable wormholes, even in classical theories. In this work
we adopt the view that the non-minimal coupling can be viewed as part of an effective field
theory (EFT) in which the field value is controlled by the theory’s cutoff. Under this assumption,
the average null energy condition, whose violation is necessary to allow traversable wormholes, is
obeyed both classically and in the context of quantum field theory. In addition, we establish a type
of “smeared” null energy condition in the non-minimally coupled theory, showing that the null
energy averaged over a region of spacetime obeys a state dependent bound, in that it depends on
the allowed field range. We finally motivate our EFT assumption by considering when the gravity
plus matter path integral remains semi-classically controlled.