# Research Publications, Xue-Mei Li

The list of my publications is given below by topics, A thesis and a dissertation are also appended at the end. These topics are:

(1) Stochastic Averaging and Homogenization, Perturbations to Conservation Laws; (2) Hypoelliptic Operators;

(3) Estimates for Fundamental Solutions of Parabolic Equations; (4) Strictly Local Martingales, Martingales, Semi-martingales;

(5) Existence of Global Smooth Stochastic Flow; (6) Complete SDEs Lacking Strong Completeness; (7) Other Stochastic Flows;

(8) Moment Estimates, Moment Stability, Long Time Asymptotics of Stochastic Dynamics, Explosion;

(9) Interplay between Stochastic Processes and their Underlying Spaces; (10) Differential Forms and Harmonic Maps;

(11) The Geometry of Diffusions; (12) Books on Diffusion Operators, Semi-elliptic Riemannian Geometry, Geometry of Filtering;

(13) Malliavin Calculus; (14) Generalised KPP Equation; (15) Spectral Gap and Poincare Inequality on Spaces of Continuous Paths;

(16) L2 Hodge-de Rham Theory in Infinite Dimensional Spaces; (17) Sobolev Calculus on Path and Loop Spaces;

## Stochastic Averaging and Homogenization, Perturbations to Conservation Laws

Problems from mathematical physics or from geometry lead to the study of singular perturbation problem. This includes dynamical descriptions for Brownian motions, convergence of manifolds, and in-homogeneously scaled Riemannian manifolds. This singular perturbation problem involves multi-scale analysis, averaging and stochastic differential equations. We study and view singular perturbation problems as perturbations to conservations laws'. Our models do not always have conventional conservation laws, e.g. real valued energy functions. Typically they are manifold valued, representing orbits. The models are motivated by Physics and Geometry. In particular we study dynamical descriptions for the convergence of a family of Riemannian manifolds to lower dimensional spaces, modelling Brownian motions as geodesics with fast changing directions, and perturbation to integrable stochastic systems which has connection to evolution of energy of weakly interactive particle systems. The analysis involving multi-scale analysis, separation of slow and fast, reduction. The stochastic parallel transport processes along the slow variables will typically also converge, in the Wasserstein distance, with rate of convergence. The state space are typically not compact. The singular operators, themselves not necessarily hypoelliptic. The fast variables are typically hypoelliptic, or more generally Fredholm operators with zero Fredholm index allowing non-unique invariant probability measures.
• Perturbation of Conservation Laws and Averaging on Manifolds (2017), article. Xue-Mei Li, Submitted to: Abelsymposium 2016.
• Homogenisation on Homogeneous Spaces (with an appendix by D. Rumynin) (2016) Article arXiv:1505.06772. To appear: Journal of the Mathematical Society of Japan. Xue-Mei Li
• Limits of Random Differential Equations on Manifolds. Article. Probability Theory Related Fields. Vol 166:659-712, No 3-4 (2016). 10.1007/s00440-015-0669-x arXiv:1501.04793, Xue-Mei Li
• Random Perturbation to the geodesic equation. The Annals of Probability , 44(1), 544-566, (2016), Xue-Mei Li. arxiv:1402.5861 article
• Effective Diffusions with Intertwined Structures, Xue-Mei Li (2012) Article Arxiv:1204.3250
• An averaging principle for Integrable stochastic Hamiltonian systems Nonlinearity 21 (2008) pp.803--822, Xue-Mei Li.

### Fredholm, Hypoelliptic Operators

In the following article we study aspects of hypoelliptic operators. Hypoelliptic operators are also studied in the following articles: Homogenisation on Homogeneous Spaces (with an appendix by D. Rumynin) (2016), Limits of Random Differential Equations on Manifolds; and Perturbation of Conservation Laws and Averaging on Manifolds (2017)Perturbation of conservation laws and averaging on manifolds (2017), arxiv 1705.08857, submitted to Abelsymposium 2016'.
• On Hypoelliptic bridge. Aticle Electronic Communications in Probability, Vol 24, no 20, 1-12 (2016), Xue-Mei Li
• Perturbation of Conservation Laws and Averaging on Manifolds (2017), article. Xue-Mei Li, submitted to Abelsymposium 2016.

### Strictly Local Martingales, Martingales, Semi-martingales

In the first and the second article we study local martingales which are not true martingales (and name them as strictly local martingales). In the third we give more examples of strict local martingale. In the fourth we study manifold valued martingales. In the fifth we show a class of hypoelliptic bridge processes are semi-martingale and has L1 integrability property.

### Brownian, Generalised and hypoelliptic Bridges

• On the Semi-Classical Brownian Bridge Measure' arxiv:1607.06498 Article Electronic Communications in Probability 2017, Vol. 22, paper no. 38, 1-15
• Generalised Brownian bridges: examples. arxiv:1612.08716 (2016), Xue-Mei Li. To appear: Markov Processes and Related Fields.

### Existence of Global Smooth Stochastic Flows (strong p-completeness)

• Strong p-completeness of stochastic differential equations and the existence of smooth flows on non-compact manifolds.  Probab. Theory Relat. Fields. 100, 4, 485-511. 1994, Xue-Mei Li.
• For this problem we assume pathwise uniqueness and local existence. An SDE is complete (or conservative) if for every initial point x, its solution F_t(x, omega) exists for almost surely all omega. An SDE is said to be strongly complete (or has a smooth global solution flow) if there exists a set omega of full measure on which F_t(x, omega) is jointly continuous. If so, the regularity of the solution will be automatically prescribed by the regularity of its vector field. If the adjoint SDE (the SDE with its drift vector field reversed) is also strongly complete then F_t(., omega) are diffeomorphisms. If the driving vector fields of and SDE are smooth, the solution is unique. These however do not imply that the solutions are continuous w.r.t. its initial data. The random solution flow needs to start with a random initial data, e.g. restart from a random variable. The technical problem is that the life times and exit times from relatively compact open sets may not be continuous with respect to the its initial value. For example the life time of the ODE (d/dt) z(t)=z(t)^2 on the complex plane is not continuous on any open set containing a segment of the real line. The solution to this equation is global if and only if the initial point is not on the real axis.

For a compact smooth manifold this is automatic (solved by lifting the SDE to the space of diffeomorphism group). For Lipschitz continuous vector fields in an Euclidean space this is solved by fixed point method. Neither method seems to extend non-compact non-linear manifolds.

Under what conditions is a complete SDE driven by smooth vector fields strongly complete? We solved this open question which was open since the 70's when the first counter example was given. This leads also to improvements to the known results on Rn, in a non-trivial way.

The strong completeness problem is roughly equivalent to the problem that for a full set of omega, a connected relatively compact open set pushed by the solution remains connected for all time.

The existence of a derivative of F_x(., -) in probability is much easier, which will solve an SDE and is called the derivative flow. We prove that the order of integration f(F_t(x, omega), where f is a smooth compactly supported function, (with respect to omega) and differentiation (in probability with respect to the initial point) can be exchanged provided the following holds: the solution with initial point from a smooth compact curve remains a connected smooth curve (We say the SDE is strongly 1-completeness).

We also introduced the concept of strong p-compeletenss, examples of SDEs which are strong p-complete not strongly (p-1) complete are also given. The main existence theorem in the following article uses the moments of the derivative flow (Thm. 4.1). We found that both the growth of the driving vector fields and their derivatives are needed, and they compensate each other. Application to SDE in Euclidean space is in section 6. Application in differentiation hear semi-group is in section 9.
• Strong completeness for a class of stochastic differential equations with irregular coefficients , X. Chen and Xue-Mei Li. Electronic J. of Probability. 19 (2014) no. 91 1-34. article. also in arXiv:1402.5079
• In this article, we separate the problems caused by the lack of regularity in the driving force from the problem that the derivatives' of the driving forces are not bounded, allowing also the ellipticity of the Markov generators decaying at infinity.

### Complete SDEs Lacking Strong Completeness

• Lack of strong completeness for stochastic flows, Xue-Mei Li and Michael Scheutzow. Annals of Probability, vol 39(4), 1407-1421 (2011).
• We give such an example on R2 with one single real valued Brownian motion and bounded smooth driving vector fields, by proving that solutions starting from two nearby points, with the same one dimensional driving noise, becomes progressively independent.

### Other Stochastic Flows

• Reflected Brownian Motion: selection, approximation, and Linearization. M. Arnaudon and Xue-Mei Li. arxiv:1602.00897 Article Electronic Journal of Probability 2017, Vol. 22, paper no. 31, 1-55
• We construct a family of approximate SDEs whose solutions stay in the interior whose damped parallel translations approximate that of a reflected Brownian motion, the latter solves the heat equation for differential 1-forms with absolute boundary conditions. This allows us to easily pass the differentiation formula from manifolds without boundary to manifolds with boundary. The difficulty, besides the non-linearity, is the convergence of continuous stochastic processes with stochastic processes with jumps. On the real line, the approximation selects the Skorohod solution versus the Tanaka solution. The damped parallel transports along the approximated paths, which are sample continuous, converge to a tangent process' along the reflected Brownian motion, with jumps upon its hitting the boundary, at the ends of the excursions of the process to the interior.
• Barycentres of probability measures transported by stochastic flows. M. Arnaudon and Xue-Mei Li. (2005) Annals of Probability, vol 33, No.4, 1509-1543.
• Given a subset of a manifold, push it with the stochastic flow of an SDE. WE study the problem how does the centre of the mass evolve.

### Interplay between Stochastic Processes and their Underlying Spaces

• On extensions of Myers' theorem. Xue-Mei Li.   Bull. London Math. Soc. 27, 392-396. 1995.
• In the article above we prove that the weighted measure exp(2h)dx is finite, hence the existence of a finite invariant probability measure for the Laplace-Beltrami operator with drift the gradient of a function h, under stochastic positivity condition on Ricci-Hess (h). This leads to a theorem on the finiteness of the fundamental group.
• Bounded and L^2 Harmonic Forms on Universal Covers.    Geom.  And Funct. Anal. Vol. 8 , 283-303 (1998). K. D. Elworthy, Xue-Mei Li and and S. Rosenberg.
• In the article above, we further explore the concept of stochastic positivity of a function. The function is question in this article are the Weitzenbock curvatures on differential forms.
• On compactness of manifolds and existence of certain type functional inequalities. Xue-Mei Li and F.-Y. Wang. Infinite Dimensional Analysis, Quantum Probability and Related Topics. Volume 6, pages 29-38, (2003).
• Curvature and topology: spectral positivity.  Methods and Applications of Global Analysis, Ed. Yu Gliklikh. In Voronezh Series on New Developments in Global Analysis.p. 45-60. 1993. K. D. Elworthy, Xue-Mei Li and Steven Rosenberg.

### The Geometry of Diffusions

Please also see the two books The Geometry of Filtering' and On the Geometry of Diffusion Operators and Stochastic Flows'.

### Books on the Geometry of Diffusion Operators and Semi-elliptic Geometry

• On the Geometry of Diffusion Operators and Stochastic Flows. Lecture Notes in Mathematics, volume 1720, Springer-Verlag (1999). K. D. Elworthy, Y. LeJan, and Xue-Mei Li. Introduction
• This is original research, not published elsewhere, on diffusion processes and their associated Riemannian and sub-Riemannian geometry. The main theorem of the book is given in the beginning of the book (page 8), the rest of book is on applications of the theorem.

We describe the main theorem. Consider a sub-elliptic operator of the form sum(X_i)^2+X_0 where X_1, ..., X_p are smooth vector fields. This a gives rise to a sub-bundle, E, of the tangent bundle. The main theorem is: a semi-elliptic second order differential operator without zero order term (diffusion operator) in Hormander form determines a metric linear connection on the sub-bundle of the tangent bundle with respect to its intrinsic metric. Furthermore every metric connection on a sub-bundle is determined this way. This connection is furthermore determined by the following property: if X is the bundle map from the trivial bundle over the manifold to the tangent bundle, the covariant derivative of the vector field, X(e), vanishes in the directions the vector fields are not trivial.

This connection has in general a torsion. For example the vector fields determined by an isometric embedding of a Riemannian manifold determines the Levi-Civita connection. We compute and study this connection: its torsion, its curvatures, Weiztenbock formulas, the symmetry property of its Ricci curvature in associated with properties of its torsion. We study also its semi-groups. A Hormander form diffusion operator of constant rank leads to an semi-elliptic stochastic differential equation (SDE) and thus probabilistic representations for various problems in analysis, especially analysis on path spaces. In particular we have Bismut type formula, Logarithmic Sobolev and Clark-Ocone formula, and decomposition of noise.
• The Geometry of Filtering, Birkhauser, (2010). K. D. Elworthy, Yves Le Jan and Xue-Mei Li.
• This is original research, not published elsewhere, on intertwined diffusion and differential operators on a principal bundle. In this book we study second order differential operators of constant rank, A and B, without zero order term, intertwined by a map from one manifold to another. The intertwining property determines a splitting of the operator B, into the sum of two Hormander type operators, which are called the horizontal operator and the vertical operator. The vertical operator can be thought of being independent' of the operator A, and hence leading to applications to stochastic filtering and to analysis on path spaces for probability measures determined by degenerate diffusion operators. This method are also useful even when the states of the operators are Euclidean spaces.

### Malliavin Calculus

Following the elementary proof for Bismut formula in my Ph.D. thesis, we explore a range of Bismut type formulae, along with their applications. See also the following articles Generalised Brownian bridges: examples', Xue-Mei Li, arxiv:1612.08716 (2016) and On Hypoelliptic bridge' Aticle  Xue-Mei Li, Electronic Communications in Probability. Vol 24, no 20, 1-12 (2016); First Order Feynman-Kac Formula. Xue-Mei Li and J. Thompson. (2016) arxiv:1608.03856 Article

### Spectral Gap and Poincare Inequality on Path Spaces

We are concerned with the differential operator d*d where the adjont is taken with respect to the probability distribution of the continuous paths of a Brownian motion or a Brownian bridge in a finite dimensional manifold. On such inequalities path spaces, for measure generated by semi-elliptic (degenerate) diffusion operators, this is also discussed in our books. Here we focus on the Brownian bridge measure on the space of continuous loops over a finite dimensional manifolds.
• A Concrete Estimate For The Weak Poincare Inequality On Loop Space. X. Chen, Xue-Mei Li, and B. Wu. Probab. Theory Relat. Fields (2011) 151:559-590
• A spectral gap for the Brownian bridge measure on hyperbolic spaces X. Chen, Xue-Mei Li, and B. Wu, Progress in analysis and its applications, 398-404, World Sci. Publ., Hackensack, NJ, 2010.
• A Poincare Inequality on Loop spaces, X. Chen, Xue-Mei Li, and B. Wu. J. of Funct. Anal. vol 259 (6), pp 1421-1442 (2010)
• ### Toward an L2 Hodge-de Rham Theory on Infinite Dimensional Spaces

The well known DeRham theorem relates DeRham cohomogolies, a differential geometric object, with singer cohomologies, a topological concept, with far reaching consequences. Toward this aim we first study the existence of an L2 Hodge decomposition theorem on path spaces, manifolds modelled on infinite dimensional Banach spaces. The difficulty is that there is no natural L2 tensor spaces of the tangent spaces.

### Toward a Sobolev Calculus on Path and Loop Spaces

The aim is to establish Sobolev calculus in infinite dimensional spaces where the derivatives are in the sense of Malliavin calculus. The idea is to establish a chart that pulls the corresponding calculus from Wiener space. One technique is using stochastic flows from an SDE whose measure would provide the basis of the analysis, and whose vector fields would induce a good covariant derivative on the finite dimensional manifold, and a stochastic filtering technique keeping all concepts intrinsic.
• Itô maps and analysis on path spaces. K. D. Elworthy and Xue-Mei Li. Math. Zeitschrift. (2007) 257: pp 643--706.
• The stochastic differential equation approach to analysis on path space. New trends in stochastic analysis and related topics, 207-226, (2012)
• Intertwining and the Markov uniqueness problem on path spaces K. D. Elworthy and Xue-Mei Li. In Stochastic Partial Differential Equations and Applications - VII, eds G. Da Prato, L. Tubaro. (2005).
• Gross-Sobolev spaces on path manifolds: uniqueness and intertwining by Itô maps. K. D. Elworthy and Xue-Mei Li. C. R. Acad. Sci. Paris, Ser. I 337 (2003) 741-744.
• Here we attempt to prove Markov uniqueness for the operator d*d. We only managed to prove that the closure of BC2 functions are the same as the closure of smooth compactly supported functions. The gap would be filled if we know the closure of BC1 functions are the same as well.

### Thesis and Dissertation

• Stochastic Flows on Noncompact manifolds (1992) In chapter 2, d+d* is proven to be essentially self-adjoint (on differential forms of all degrees), d* being the L2 adjoint of d w.r.t the measure exp(2h)dx, h a smooth function, dx the Riemmannian volume measure, followed by a Hodge decomposition theorem, commutation of d with the corresponding semi-group, a discussion on recurrent property (chapter 3) and invariant measure. In chapter 4, an over determined heat equation is considered together with a criterion for non-explosion, non inplosion, and C_0 property. In Chapter 5, the open problem for the existence of a global solution flow of an SDE on non-compact manifold is solved. In chapter 6, we define \delta P_t phi(v)=E df (TF_t(v)), where TF_t(v) is the derivative flow of the Brownian flow with drift nabla h, phi a differential 1-form. The questions weather dP_tf=\delta P_t df and \delta P_t df=e^{1/2t]\Delta}df, where f is a smooth function, are studied. differentiation under the expectation is also studied. These are important for the next chapter. In chapter 7, moment stability of Brownian system are studied together with the topological properties of the manifolds, relating to vanishing of harmonic differential forms and cohomologies. In chapter 8, A Bismut type formula are extended to differential forms of degree greater than one, employing the gradient Brownian system. As a by product, a simple elementary proof for Bismut’s formula (for functions ) is given.
• Behaviour at infinity of stochastic solutions and diffusion semi-groups (1989), M.Sc. dissertation.