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 within the scope of stochastic analysis, rough path theory and their applications, including (but not restricted to):

Rough paths, the signature transform
Stochastic/Rough differential equations, diffusion processes
Gaussian processes, the Malliavin calculus
Stochastic analysis on manifolds

Preprints and publications:

On the lack of Gaussian tail for rough line integrals along fractional Brownian paths (with H. Boedihardjo), preprint, 2022.
Non-degeneracy of stochastic line integrals (with S. Wang), preprint, 2022.
When is the Conway-Maxwell-Poisson distribution infinitely divisible? (with A. Xia), Stat. Probab. Lett. 181, 2022.
Precise local estimates for differential equations driven by fractional Brownian motion: hypoelliptic case (with C. Ouyang and S. Tindel), Ann. Probab. 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 J. Math. 59 (3): 653-682, 2022.
SL_2(R)-developments and signature asymptotics for planar paths with bounded variation (with H. Boedihardjo), Rev. Mat. Iberoam. 2022.
Precise local estimates for differential equations driven by fractional Brownian motion: elliptic case (with C. Ouyang and S. Tindel), J. Theoret. Probab. 2022.
Long time asymptotics of heat kernels and Brownian winding numbers on manifolds with boundary (with G. Iyer), Lat. Am. J. Probab. Math. Stat. 18: 1297-1323, 2021.
Path developments and tail asymptotics of signature for pure rough paths (with H. Boedihardjo and N. Souris), Adv. Math. 364, 2020.
Tail asymptotics of the Brownian signature (with H. Boedihardjo), Trans. Amer. Math. Soc. 372: 585-614, 2019.
A non-vanishing property for the signature of a path (with H. Boedihardjo), C. R. Math. Acad. Sci. Paris 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 Anal. 51 (1): 1-21, 2019.
Reconstruction for the signature of a rough path, Proc. Lond. Math. Soc. 114 (3): 495-526, 2017.
Finite dimensional characteristic functions of the Brownian rough path (with Z. Qian), Front. Math. China 12 (4): 859-877, 2017.
The signature of a rough path: uniqueness (with H. Boedihardjo, T. Lyons and D. Yang), Adv. Math. 293: 720-737, 2016.
On an inversion theorem for Stratonovich's signatures of multidimensional diffusion paths (with Z. Qian), Ann. Inst. Henri Poincaré Probab. Stat. 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 J. Math. 53 (4): 941-970, 2016.
The uniqueness of signature problem in the non-Markov setting (with H. Boedihardjo), Stochastic Process. Appl. 125 (12): 4674-4701, 2015.
Simple piecewise geodesic interpolation of simple and Jordan curves with applications (with H. Boedihardjo), Constr. Approx. 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.