## Status:

## Personal website:

+44 1865 615172

## Research groups:

## Address

University of Oxford

Andrew Wiles Building

Radcliffe Observatory Quarter

Woodstock Road

Oxford

OX2 6GG

## Research interests:

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.

## Preferred address:

isao.sauzedde[at]maths.ox.ac.uk

## Major / recent publications:

https://arxiv.org/abs/2101.03992

**Lévy area without approximation**. 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.

https://arxiv.org/abs/2102.12372

**Planar Brownian motion winds evenly along its trajectory**. 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.

https://arxiv.org/abs/2105.01232

**Integration and stochastic integration in Gaussian multiplicative chaos**. 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.