Xi Geng's Homepage

Research Interests:

I am a mathematician working in the interplay between probability and analysis. I particularly enjoy and appreciate the deeper intrinsic connections among different mathematical areas. My research interests are surrounding the study of rough path theory, the signature transform, quantitative aspects of rough differential equations and SPDEs, stochastic analysis on manifolds. Recently, I am particularly interested in understanding how large scale behaviours of random dynamics are related to the global geometry in various geometric settings.

Preprints and publications:

Hitting probabilities for hypoelliptic differential equations driven by fractional Brownian motion (with S. Wang), arXiv preprint, 2025.
Parabolic Anderson model in the hyperbolic space. Part II: quenched asymptotics (with S. Wang and W. Xu), arXiv preprint, 2025.
Parabolic Anderson model in the hyperbolic space. Part I: annealed asymptotics (with W. Xu), arXiv preprint, 2025.
Cartan's path development, the logarithmic signature and a conjecture of Lyons-Sidorova (with H. Boedihardjo and S. Wang), arXiv preprint, 2025.
Parabolic Anderson model in hyperbolic spaces and phase transition (with C. Ouyang), arXiv preprint, 2025.
A coupling between random walks in random environments and Brox's diffusion (with M. Gradinaru and S. Tindel), arXiv preprint, 2024.
Expected signature on a Riemannian manifold and its geometric implications (with H. Ni and C. Wang), arXiv preprint, 2024.
Stochastic domination of exit times for random walks and Brownian motion with drift (with G. Markowsky), Stochastic Models, 2025.
On the lack of Gaussian tail for rough line integrals along fractional Brownian paths (with H. Boedihardjo), Probability Theory and Related Fields 188: 1287-1313, 2024.
Non-degeneracy of stochastic line integrals (with S. Wang), Electronic Journal of Probability 28: 1-28, 2023.
SL_2(R)-developments and signature asymptotics for planar paths with bounded variation (with H. Boedihardjo), Revista Matemática Iberoamericana 39 (5): 1973-2006, 2023.
Precise local estimates for differential equations driven by fractional Brownian motion: elliptic case (with C. Ouyang and S. Tindel), Journal of Theoretical Probability 36: 1341-1367, 2023.
When is the Conway-Maxwell-Poisson distribution infinitely divisible? (with A. Xia), Statistics and Probability Letters 181, 2022.
Precise local estimates for differential equations driven by fractional Brownian motion: hypoelliptic case (with C. Ouyang and S. Tindel), The Annals of Probability 50 (2): 649-687, 2022. The original version of this article can be found here.
Lipschitz-stability of controlled rough paths and rough differential equations (with H. Boedihardjo), Osaka Journal of Mathematics 59 (3): 653-682, 2022.
Long time asymptotics of heat kernels and Brownian winding numbers on manifolds with boundary (with G. Iyer), Latin American Journal of Probability and Mathematical Statistics 18: 1297-1323, 2021.
Path developments and tail asymptotics of signature for pure rough paths (with H. Boedihardjo and N. Souris), Advances in Mathematics 364, 2020.
Tail asymptotics of the Brownian signature (with H. Boedihardjo), Transactions of the American Mathematical Society 372: 585-614, 2019.
A non-vanishing property for the signature of a path (with H. Boedihardjo), Comptes Rendus Mathematique 357 (2): 120-129, 2019.
A quasi-sure non-degeneracy property for the Brownian rough path (with H. Boedihardjo, X. Liu and Z. Qian), Potential Analysis 51 (1): 1-21, 2019.
Reconstruction for the signature of a rough path, Proceedings of the London Mathematical Society 114 (3): 495-526, 2017.
Finite dimensional characteristic functions of the Brownian rough path (with Z. Qian), Frontier of Mathematics in China 12 (4): 859-877, 2017.
The signature of a rough path: uniqueness (with H. Boedihardjo, T. Lyons and D. Yang), Advances in Mathematics 293: 720-737, 2016.
On an inversion theorem for Stratonovich's signatures of multidimensional diffusion paths (with Z. Qian), Annales de l'institut Henri Poincare (B) Probability and Statistics 52 (1): 429-447, 2016.
Quasi-sure existence of Gaussian rough paths and large deviation principles for capacities (with H. Boedihardjo and Z. Qian), Osaka Journal of Mathematics 53 (4): 941-970, 2016.
The uniqueness of signature problem in the non-Markov setting (with H. Boedihardjo), Stochastic Processes and Their Applications 125 (12): 4674-4701, 2015.
Simple piecewise geodesic interpolation of simple and Jordan curves with applications (with H. Boedihardjo), Constructive Approximation 42 (1): 161-180, 2015.
G-Brownian motion as rough paths and differential equations driven by G-Brownian motion (with Z. Qian and D. Yang), Séminaire de Probabilités XLVI, Lecture Notes in Mathematics, 125-193, 2014.