# Isao Sauzedde

**Address**

Mathematical Institute

University of Oxford

Andrew Wiles Building

Radcliffe Observatory Quarter

Woodstock Road

Oxford

OX2 6GG

**Lévy area without approximation**. https://arxiv.org/abs/2101.03992 (to be published in Annales de l'Institut Henri Poincaré (B) )

An almost sure Green formula for the planar Brownian motion is proven, by studying the area of the sets of points $D_N$ with large Brownian winding. The average winding between a planar Brownian motion and a Poisson point process of large intensity on the plane is also studied.

This conduces to a more geometric interpretation of the Lévy area, which does not rely on approximations of the Brownian path, or on the Euclidean structure of the plane.

**Planar Brownian motion winds evenly along its trajectory**. https://arxiv.org/abs/2102.12372

The sets $D_N$ of points with Brownian winding at least $N$ is studied further. The measures $2\pi N \mathbf{1}_{D_N}$ associated with the sets $D_N$ are proved to converge almost surely weakly toward the occupation measure of the Brownian motion.

**Winding and intersection of Brownian motions**.** **https://arxiv.org/abs/2112.01645

Most of the results of the two article above are extended to the set of points $D_{n,m}$ around which two independent Brownian motions both wind a lot of time (at least $n$ and $m$ times respectively). The asymptotic area of this set is identifed, together with an upper bound on the convergence rate, both in the $L^p$ and almost sure sense. The Lebesgue measure carried by $D_{n,m}$, normalised by $4\pi^2 nm$, is shown to converge almost surely weakly (and also in $L^p$ for the Wasserstein $1$-distance) toward the intersection measure of the Brownian motions.

**Integration and stochastic integration in Gaussian multiplicative chaos**. https://arxiv.org/abs/2105.01232

We use our previous approach of *Lévy area without approximation *to prove that it is possible to define the Lévy area of the Brownian motion, when the underlying Lebesgue area measure is replaced with an highly singular area measure provided that this area measure is itself random.

The regularity assumptions on the measure that are imposed by the classical stochastic calculus can then be trade with regularity assumptions on the covariance kernel of the measure. The paper deals specifically with the case of the Gaussian multiplicative chaos, but the construction extends beyond this framework. Some regularity properties of the integral so defined are also obtained.

**Homotopy and holonomy of the planar Brownian motion in a Poisson punctured plane**. https://arxiv.org/abs/2110.15816

We define a family of diffeomorphism-invariant models of random connections on principal G-bundles over the plane, whose curvatures are concentrated on singular points. In a limit when the number of point grows whilst the singular curvature on each point diminishes, the model converges in some sense towards a Yang--Mills field. We study another regime for which we prove that the holonomy along a Brownian trajectory converges towards an explicit limit.

Planar Brownian motion, rough paths, random matrix, differential geometry and Gauge fields are my primary interests for the time being. I am also very interested in SPDE, though my knowledge on the subject is rather sparse.

During my PhD, I have studied the windings of the planar Brownian motion, in order in particular to get a coordinate-free definition of stochastic integration. In the future, I would like to extend this study to other processes, including fractional Brownian motion. From a rough path point of view, my goal is basically to define geometrically significant rough path extensions of curves.

Currently, I am looking at the linear statistics of time-dependent large unitary matrices, Haar distributed and with Brownian evolution.

As a postgraduate, I have studied the stochastic proof of Chern-Gauss-Bonnet theorem, and I remain very interested with probabilistic interpretations of topological invariants.