## Summary

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 applied mathematical models such as geometric fluid equations. See LIST OF PUBLICATIONS. The List of Publication by Topics might be more informative.

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 at here.

## Selected Publications

### Journal Articles

ELWORTHY KD, LI XM, 1994, FORMULAS FOR THE DERIVATIVES OF HEAT SEMIGROUPS, *Journal of Functional Analysis*, Vol:125, ISSN:0022-1236, Pages:252-286

LI XM, 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 XM, 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 XM, 1995, ON EXTENSIONS OF MYERS THEOREM, *Bulletin of the London Mathematical Society*, Vol:27, ISSN:0024-6093, Pages:392-396

Li XM, 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

Elworthy KD, Li XM, 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

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 XM, Rosenberg S, 1998, Bounded and L-2 harmonic forms on universal covers, *Geometric and Functional Analysis*, Vol:8, ISSN:1016-443X, Pages:283-303

Arnaudon M, Li XM, 2005, Barycenters of measures transported by stochastic flows, *Annals of Probability*, Vol:33, ISSN:0091-1798, Pages:1509-1543

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

Chen X, Li X-M, Wu B, 2010, A Poincare inequality on loop spaces, *Journal of Functional Analysis*, Vol:259, ISSN:0022-1236, Pages:1421-1442

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, 2011, A concrete estimate for the weak Poincar, inequality on loop space, *Probability Theory and Related Fields*, Vol:151, ISSN:0178-8051, Pages:559-590

Li X-M, Hessian formulas and estimates for parabolic Schrödinger operators

Li X-M, Perturbation of Conservation Laws and Averaging on Manifolds

Li X-M, 2016, On hypoelliptic bridge, *Electronic Communications in Probability*, Vol:21, ISSN:1083-589X

Arnaudon M, Li X-M, 2017, Reflected Brownian motion: selection, approximation and linearization, *Electronic Journal of Probability*, Vol:22, ISSN:1083-6489

Li X-M, 2016, RANDOM PERTURBATION TO THE GEODESIC EQUATION, *Annals of Probability*, Vol:44, ISSN:0091-1798, Pages:544-566

Li X-M, 2016, Limits of random differential equations on manifolds, *Probability Theory and Related Fields*, Vol:166, ISSN:0178-8051, Pages:659-712

### Books

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, LeJan Y, Li X-M, *The Geometry of Filtering*

### 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