Introduction to Regularity Structures

Abstract

These are short notes from a series of lectures given at the University of Rennes in June 2013, at the University of Bonn in July 2013, at the XVII th Brazilian School of Probability in Mambucaba in August 2013, and at ETH Zurich in September 2013. They give a concise overview of the theory of regularity structures as exposed in the article [Hai14]. In order to allow to focus on the conceptual aspects of the theory, many proofs are omitted and statements are simplified. We focus on applying the theory to the problem of giving a solution theory to the stochastic quantisation equations for the Euclidean Φ 4 3 quantum field theory.

Learning Paths from Signature Tensors

“…Consider the m 2 × m 2 submatrix J 1 of the Jacobian obtained by setting k = 1 in (12). The entry of J 1 in row (i, j) and column (u, v) is the linear form The same conclusion holds if any of the maximal minors of the Jacobian in K m k ×m 2 is non-zero.…”

“…Its entry q ij is the Lévy area of the projection of ψ onto the plane indexed by i and j, the signed area between the planar path and the segment connecting its endpoints. For background on signature tensors of paths and their applications see [1,8,10,12,21,22,23].…”

“…The path is encoded by its signature, an infinite sequence of tensors that are interrelated through a Lie algebra structure. Signature tensors were introduced by Chen [7], and they play an important role in stochastic analysis [12,21]. We refer to [22,23] for the recovery problem, and to [10,13,17,18] for algorithms and applications.…”

Varieties of Signature Tensors

“…They are central to the theory of rough paths [30], a revolutionary view on stochastic analysis, with important contributions by Terry Lyons, Martin Hairer and many others (cf. [19,21,32,33]).…”

A rough calculus approach to level sets in the Heisenberg group

“…At first glance, the LSDE is similar to equations driven by a rough signal appearing in the theory of rough paths as exposed, for example, in [9,12,17], but it is different, being inherently 'autonomous', while the usual rough differential equations (RDEs) are not. However, the theory of rough paths still provides an appropriate tool, namely, the sewing lemma, to construct solutions to LSDEs, thus enabling some 'differential' calculus for maps on the Heisenberg group, regular only in the intrinsic sense of the latter, but possibly nowhere differentiable on a set of positive measure [18].…”