## Summary

I work in Probability, also interested in Analysis and Differential Geometry. I have always been interested in the evolution of physical quantities approximated by solutions of differential equations, 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. PUBLICATIONS. List of Publication by Topics (not kept up to date). Recent preprints are on the arXiv.

Before joining Imperial, I was a professor at the University of Warwick and worked in the USA, at 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, and a lecturer in a number intermantional schools.

I serve as an associate editor of the Electronic Journal of Probability and the Electronic Communications in Probability. I am an organiser of `Geometry, Stochastic, & Dynamics', a co-investigator of the EPSRC grant EP/S023925/1 and in the management committee of the CDT in Mathematics of Random Systems. This past year, I gave a plenary lecture at the triannual International Congress of Chinese Mathematicians, was an organiser and in the program committee for the International Program on Regularity Structures and Stochastic Systems (see programme), and in the Scientific Committee for Stochastic Processes and their Applications 2019. I love to promote public awareness of science and mathematics and support young and under presented groups in science and technology. I gave talks at 2019 LMS Women in Mathematics kent, Cambridge student chapter SIAM-IMA 2019, and 50th Anniversary of Mathematics in La laguna University. At Imperial college London I gave talks at `Love of Math' and `Mary Lister McCammon Summer Research Program'.

My personal journey in mathematics began when I entered Hebei University as an undergraduate, at the age of 15. Upon graduation I briefly took up teaching, before entering the graduate school of Xian Jiaotong University. I was then awarded a 4-year Sino-British Friendship Scholarship from the Ministry fo higher education, for studying at the U.K. I then studied and obtained my Ph.D. in mathematics at the University of Warwick. Before that I studied briefly at Beijing Language Institute. Here is my Personal Web page.

**Books I co-authored: **The Geometry fo Filtering

On the geometry of diffusion operators and stochastic flows

## Selected Publications

### Journal Articles

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, ISSN:0025-5645, Pages:519-572

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

### Books

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

### Chapters

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