I am a mathematician, working in the intersection of Probability, Analysis, and Differential Geometry. I have always been interested in the evolution of physical quantities approximated by solutions of differential equations, with special interests in the intrinsic geometry of stochastic (random) processes and lately use this understanding for complexity reduction and for multi scale and asymptotic analysis. The statistical properties of those stochastic processes can be described by second order differential operators and parabolic equations. I work also on gradient and Hessian estimates for the solutions of the latter. Currently I am also drawn to perturbation models, homogenisation problems, slow/fast systems, and applied mathematical models. See LIST OF PUBLICATIONS. The List of Publication by Topics might be more informative. Recent preprints can be found on-arXiv.
Before joining Imperial, I was professor at the University of Warwick and also worked in the United States (in the University of Connecticut, first as an assistant professor and then tenured and an associate professor), at MSRI Berkeley (as an NSF funded MSRI research Fellow) and in Germany (at the University of Boon and the University of Bochum as an Alexander von Humboldt Research Fellow). While at the Connecticut, I obtained consecutively three standard NSF grants (funded by the Probability Program and by the Geometric Analysis Program). I was a Royal Society Senior Leverhulme Research Fellow and a visiting professor to a number of universities / institutions.
I began to study mathematics at Hebei University at the age of 15. Upon graduation I took up the teaching profession, teaching in a senior high school. Later I entered the graduate school at Xian Jiaotong University, during which period I was awarded a full scholarship from the Sino-British Friendship Scholarship Scheme for studying in the U.K.. After a short training at the Beijing Language Institute, studying English, I came to England and studied at the Mathematics Institute at the University of Warwick. My personal webpage can be found here.
Currently, I am a co-investigator of the EPSRC grant EP/S023925/1. The grant supports a Doctoral Training Centre in Mathematics of Random Systems.
- Recently, I gave a plenary lecture at the 9th International Congress of Chinese Mathematicians.
- I like to promote public awareness of science and mathematics and support young and under presented groups in science and technology. I gave a number of talks in public events for women in mathematics as well as events for undergraduates/postgraduate students.
Research Books:` The Geometry fo Filtering', Lecture Notes in Maths,`On the geometry of diffusion operators and stochastic flows', Birkhauser. Both books are written in collaboration with Y. LeJan and K. D. Elworthy.
Li X-M, 2018, Perturbation of Conservation Laws and Averaging on Manifolds, Computation and Combinatories in Dynamics, Stochastic and Control, the Abel Symposium, Rosendal, Norway, August 2016
Li X-M, 2018, Homogenisation on homogeneous spaces, Journal of the Mathematical Society of Japan, Vol:70, Number 2, 519-572, ISSN:0025-5645
Arnaudon M, Li X-M, 2017, Reflected Brownian motion: selection, approximation and linearization, Electronic Journal of Probability, Vol:22, ISSN:1083-6489, Pages:1-55
Li X-M, 2016, Random perturbation to the geodesic equation, Annals of Probability, Vol:44, ISSN:0091-1798, Pages:544-566
Li X-M, 2015, Limits of random differential equations on manifolds, Probability Theory and Related Fields, Vol:166, ISSN:1432-2064, Pages:659-712
Li X-M, Scheutzow M, 2011, Lack of strong completeness for stochastic flows, Annals of Probability, Vol:39, ISSN:0091-1798, Pages:1407-1421
Chen X, Li X-M, Wu B, 2010, A concrete estimate for the weak Poincaré inequality on loop space, Probability Theory and Related Fields, Vol:151, ISSN:1432-2064, Pages:559-590
Chen X, Li X-M, Wu B, 2010, A Poincaré inequality on loop spaces, Journal of Functional Analysis, Vol:259, ISSN:0022-1236, Pages:1421-1442
Li X-M, 2008, An averaging principle for a completely integrable stochastic Hamiltonian system, Nonlinearity, Vol:21, ISSN:0951-7715, Pages:803-822
Elworthy KD, Li X-M, 2008, An $L^2$ theory for differential forms on path spaces. I, Journal of Functional Analysis, Vol:254, ISSN:0022-1236, Pages:196-245
Elworthy KD, Li X-M, 2007, Itô maps and analysis on path spaces, Mathematische Zeitschrift, Vol:257, ISSN:0025-5874, Pages:643-706
Arnaudon M, Li X-M, 2005, Barycenters of measures transported by stochastic flows, Annals of Probability, Vol:33, ISSN:0091-1798, Pages:1509-1543
Elworthy KD, Li X-M, Yor M, 1999, The importance of strictly local martingales; applications to radial Ornstein-Uhlenbeck processes, Probability Theory and Related Fields, Vol:115, ISSN:0178-8051, Pages:325-355
Elworthy KD, Li X-M, Rosenberg S, 1998, Bounded and $L^2$ harmonic forms on universal covers, Geometric and Functional Analysis, Vol:8, ISSN:1016-443X, Pages:283-303
Elworthy KD, Li X-M, 1998, Bismut type formulae for differential forms, Academie des Sciences. Comptes Rendus. Serie 1. Mathematique (France), Vol:327, ISSN:0764-4442, Pages:87-92
Li X-M, Zhao HZ, 1996, Gradient estimates and the smooth convergence of approximate travelling waves for reaction-diffusion equations, Nonlinearity, Vol:9, ISSN:0951-7715, Pages:459-477
Li X-M, 1995, On extensions of Myers’ theorem, Bulletin of the London Mathematical Society, Vol:27, ISSN:0024-6093, Pages:392-396
Elworthy KD, Li X-M, 1994, Formulae for the derivatives of heat semigroups, Journal of Functional Analysis, Vol:125, ISSN:0022-1236, Pages:252-286
Li X-M, 1994, Strong $p$-completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds, Probability Theory and Related Fields, Vol:100, ISSN:0178-8051, Pages:485-511
Li X-M, 1994, Properties at infinity of diffusion semigroups and stochastic flows via weak uniform covers, Potential Analysis, Vol:3, ISSN:0926-2601, Pages:339-357
Li X-M, 1994, Stochastic differential equations on noncompact manifolds: moment stability and its topological consequences, Probability Theory and Related Fields, Vol:100, ISSN:0178-8051, Pages:417-428
Li X-M, Hessian formulas and estimates for parabolic Schrödinger operators
Elworthy KD, Le Jan Y, Li X-M, 2010, The geometry of filtering, Birkhäuser Verlag, Basel, ISBN:978-3-0346-0175-7
Elworthy KD, Le Jan Y, Li X-M, 1999, On the geometry of diffusion operators and stochastic flows, Springer-Verlag, Berlin, ISBN:3-540-66708-3
Elworthy KD, Li X-M, 2006, Geometric stochastic analysis on path spaces, International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, Pages:575-594
Elworthy KD, Le Jan Y, Li X-M, 2004, Equivariant diffusions on principal bundles, Stochastic analysis and related topics in Kyoto, Math. Soc. Japan, Tokyo, Pages:31-47
Elworthy KD, Li X-M, 2000, Special Itô maps and an $L^2$ Hodge theory for one forms on path spaces, Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999), Amer. Math. Soc., Providence, RI, Pages:145-162
Elworthy KD, Li XM, Yor M, 1997, On the tails of the supremum and the quadratic variation of strictly local martingales, Séminaire de Probabilités, XXXI, Springer, Berlin, Pages:113-125
Elworthy KD, Le Jan Y, Li X-M, 1997, Concerning the geometry of stochastic differential equations and stochastic flows, New Trends in Stochastic Analysis: Proceedings of a Tanaguchi International Workshop Charingworth Manor September 21-27 1994, World Scientific, Pages:107-131
Elworthy KD, Li X-M, 1996, A class of integration by parts formulae in stochastic analysis. I, Itô’s stochastic calculus and probability theory, Springer, Tokyo, Pages:15-30
Elworthy KD, Li X-M, 1995, Derivative flows of stochastic differential equations: moment exponents and geometric properties, Stochastic analysis (Ithaca, NY, 1993), Amer. Math. Soc., Providence, RI, Pages:565-574
Elworthy KD, Li X-M, 1994, Differentiation of heat semigroups and applications, Probability theory and mathematical statistics: Proceeding of the Sixth Vilnius Conference (1993), Editor(s): Kubilius, Grigelionis, Pragarauskas, Statulevicius, TEV, Vilnius, Pages:239-251