Zu Chongzhi Mathematics Research Seminar

Date and Time (China standard time): Friday, November 17, 10:00-11:00 am

Location: WDR 1007

Zoom: 952 7021 8931; Passcode: dkumath

Title: From motion of Brownian particle in magnetic field to machine learning: PDE approaches to signature of rough path

Speaker: Siran Li

Bio: Siran Li is an associate professor at Shanghai Jiao Tong University. His main research interest lies in analysis of geometric/fluid dynamical PDEs and calculus of variations. He obtained a D.Phil. degree from the University of Oxford in 2017 under the supervision of Prof. Gui-Qiang Chen, and worked as a postdoctoral instructor with Prof. Robert Hardt at Rice University. Prior to joining Shanghai Jiao Tong University, he was a visiting assistant/adjunct associate professor at NYU Shanghai. 


The rough path theory provides a far-reaching generalisation to the well-posedness theory for controlled/stochastic differential equations. Various fundamental problems pertaining to the signature of rough paths, which is a non-commutative analogue of polynomials and whose expectation plays a role similar to that of the moment generating function, remain widely open despite the exciting current developments in rough path analysis. In this talk, we present our recent works on the expected signature of physical Brownian motion in magnetic fields and mathematical Brownian motion stopped upon its first exit time on $C^{2,\alpha}$-Euclidean domains. We shall also briefly discuss our development of a GAN model in machine learning, known as the PCF-GAN (PCF for probability characteristic function), based on theoretical studies of the signature. Throughout these works, we emphasise a novel approach via the analysis of a graded system of parabolic PDEs in the tensor algebra. (*Joint work with Prof. Hao Ni from University College London and Qianyu Zhu from MIT).