1��<!DOCTYPE html>�<html lang="en">�<head>� <meta charset="UTF-8" />� <title>Xue-Mei Li Bio</title>� <style>� body {� font-family: "Georgia", "Times New Roman", serif;� line-height: 1.6;� max-width: 800px;� margin: 40px auto;� padding: 0 20px;� color: #333;� background-color: #fafafa;� }� h1 {� font-family: "Helvetica Neue", Helvetica, Arial, sans-serif;� font-weight: 500;� text-align: center;� margin-bottom: 40px;� }� p {� font-size: 1.1em;� }� </style>�</head>�<body>�� <h1>Xue-Mei Li</h1>�� <p>� Xue-Mei Liâs work lies at the vibrant intersection of probability theory, stochastic analysis, differential geometry (both finite and infinite dimensional), and multiscale dynamics, including fractional and non-Markovian systems. Combining deep geometric insights with rigorous analysis, she explores how noise âfeelsâ curvature and topology, revealing how stochastic flows encode geometric and topological features and deepening our understanding of how randomness interacts with geometry within complex structures. She has forged remarkable collaborations, advancing the field through innovative applications of rough path theory to extend classical results to multiscale SDEs driven by rough or long-range dependent noise such as fractional Brownian motion and Volterra processes. By combining rough path theory with Malliavin calculus, she has extended large-scale dynamics and fluctuation theory for SPDEs to spatially long-range systems.� </p>�� <p>� Among her many contributions are solving the longstanding open problem of strong completeness for SDEs by establishing fundamental criteria guaranteeing global smooth flows on non-compact manifolds; proving the derivative (BEL) formula for diffusion semigroups using an innovative martingale method; and coining and pioneering the study of strict local martingales. Her work in infinite-dimensional analysis and Malliavin calculus on manifolds includes profound contributions to the analysis of degenerate diffusion measures, proving Poincaré inequalities for loop measures, and obtaining sharp logarithmic estimates for heat kernels.� </p>��While on the topic of rough path, a recent progress is:�<center>� <h2 class="title">Rough Differential Equations</h2>�</center>��<li>� <a class="addressbg" href="https://arxiv.org/abs/2502.08799">� Strong Completeness of SDEs and Non-Explosion for RDEs with Coefficients Having Unbounded Derivatives� </a>, by Xue-Mei Li and Kexing Ying (2025). In this work, they establish a non-explosion result for rough differential equations (RDEs) where both the noise and drift coefficients, together with their derivatives, are allowed to grow at infinity. This extends beyond the standard assumptions in RDE theory, which typically require the driving vector fields and their derivatives up to a certain order to be bounded to ensure global well-posedness. Additionally, they prove the existence of a bi-continuous solution flow for stochastic differential equations (SDEs). In the case of RDEs with additive noise, they show that their result is optimal by providing a counterexample.�</li>���<center>� <h2 class="title">Coarse Curvatures</h2>�</center>��<li>� <a class="addressbg" href="https://www.ams.org/journals/tran/0000-000-00/S0002-9947-2025-09482-0/?active=current">� Coarse Ricci Curvature of Weighted Riemannian Manifolds (2023)� </a>, in Transactions of the AMS (2025). In this work, M. Arnaudon, Xue-Mei Li, and Benedikt Petico show how to recover the generalized Ricci curvature of a weighted Riemannian manifold using optimal transport techniques, inspired by Ollivier's notion of coarse Ricci curvature. Specifically, they prove that the generalized Ricci tensor, in the sense of Bakry and Emery, can be obtained asymptotically by examining how Wasserstein-1 distances between locally supported, normalized weighted volume measures change under small shifts in the manifold. As an application, they study random geometric graphs sampled from Poisson point processes with non-uniform intensity, demonstrating that their limiting coarse Ricci curvature converges to the manifold's generalized Ricci tensor. This bridges discrete curvature notions and smooth geometry via optimal transport and random sampling.� <br><br>� <a class="addressbg" href="https://link.springer.com/article/10.1007/s40879-025-00816-x">� Coarse Extrinsic Curvature of Riemannian Submanifolds� </a>, in European Journal of Mathematics (2025). Following the previous work, in this article, they introduce a novel concept of coarse extrinsic curvature for Riemannian submanifolds. This curvature is derived from Wasserstein-1 distances between probability measures supported in tubular neighbourhoods of a submanifold, providing new insights into the extrinsic geometry of isometrically embedded manifolds in Euclidean spaces. Their framework also offers methods to approximate mean curvature from statistical data, such as point clouds generated by Poisson point processes. This approach has potential applications in manifold learning and metric embedding theory, enabling geometric information to be inferred directly from empirical data.� <br><br>� Together, these works establish a framework for understanding both intrinsic and extrinsic geometric quantities through coarse curvature concepts, optimal transport, and random sampling, offering new tools for geometric analysis, data science, and metric measure geometry.�</li>��<center>� <h2 class="title">Fluctuations of SHE and KPZ Equations with Long Range Dependent Noise</h2>�</center>��<li>� In these works, Xue_Mei Li et al study the large-scale behaviour of nonlinear stochastic heat equations (SHE) and the KPZ equation in dimensions \( d \ge 3 \) driven by multiplicative Gaussian noise that is white in time and spatially coloured with nonâintegrable covariance decaying as \( |x|^{-\kappa} \) for \( \kappa \in (2,d) \). They show that, unlike the case with compactly supported spatial correlations, the long-range correlations persist in the scaling limit.� <br><br>� For the stochastic heat equation, presented in � <a class="addressbg" href="https://projecteuclid.org/journals/annals-of-applied-probability/volume-35/issue-2/Fluctuations-of-stochastic-PDEs-with-long-range-correlations/10.1214/24-AAP2140.short">� Fluctuations of stochastic PDEs with long-range correlations� </a>, Annals of Applied Probability (2025), they prove that fluctuations of the diffusively rescaled solution converge to those of an additive SHE driven by noise with Riesz-kernel covariance of degree \(-\kappa\), with convergence holding in optimal Hölder topologies as distribution-valued processes.� <br><br>� For the KPZ equation (also in the same article), they demonstrate that its scaling limit is described by an additive SHE retaining the same spatial correlation. Remarkably, the limiting noise is the scaling limit of the original noise itself, leading to convergence in probability under suitable coupling. This contrasts with the integrable correlation case, where the limiting noise becomes spatially white and only weak convergence is obtained.� <br><br>� These results highlight how non-integrable, long-range spatial correlations fundamentally alter the large-scale limits of stochastic PDEs, preserving noise correlations and yielding new types of universality classes.�</li>��<!-- MathJax for rendering math -->�
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1������ <center>� <h2 class="title">Multi-Scale Analysis</h2>�</center>��<li>� Xue-Mei Li has made significant and original contributions to the theory of multi-scale stochastic systems, particularly in the context of geometric and manifold-valued models where classical methods do not apply directly. Her work combines rigorous stochastic analysis, differential geometry, and dynamical systems to address singular perturbation problems arising from physics and geometry.�� <br><br>�� A distinctive feature of her approach is viewing singular perturbations as perturbations to conservation laws, even in settings where conventional scalar conserved quantities do not exist. Instead, the conserved quantities in her models are often manifold-valued, representing orbits or geometric structures. This perspective extends classical averaging and homogenisation theory to systems with geometric constraints, integrable structures, and non-trivial topology.�� <br><br>�� <strong>Key contributions include:</strong>�� <ul>� <li>� <strong>Stochastic averaging and homogenisation on manifolds.</strong>� She developed general frameworks for stochastic averaging and homogenisation when the state space is a manifold or a homogeneous space"", such as in � <a class="addressbg" href="https://arxiv.org/abs/1505.06772">� "Homogenisation on Homogeneous Spaces (with an appendix by D. Rumynin) (2016)</a>. These results generalise classical Euclidean homogenisation by incorporating geometric structures and group actions, making them applicable to geometric flows and mechanical systems with symmetries.� </li>�� <li>� <strong>Perturbations of geodesic equations.</strong>� In � <a class="addressbg" href="http://www2.warwick.ac.uk/fac/sci/maths/people/staff/xue_mei_li/geodesic.pdf">� "Random Perturbation to the Geodesic Equation" (2016)</a>, Li models Brownian motion as a geodesic flow with rapidly changing directions, providing a stochastic dynamical description of Brownian motion as the scaling limit of perturbed geodesic flows. This creates a bridge between deterministic Hamiltonian dynamics and stochastic processes on manifolds.� </li>�� <li>� <strong>Limits of random differential equations on manifolds.</strong>� Her paper � <a href="http://link.springer.com/article/10.1007%2Fs00440-015-0669-x"> "Limits of Random Differential Equations on Manifolds" (2016)</a> systematically analyses random differential equations under multi-scale perturbations, establishing convergence results for the slow variables and associated stochastic parallel transport, often with convergence rates in Wasserstein distance, even when the state space is non-compact and the operators are not hypoelliptic.� </li>�� <li>� <strong>Perturbation of conservation laws and averaging on manifolds.</strong>� In � <a class="addressbg" href="https://arxiv.org/abs/1705.08857">"Perturbation of Conservation Laws and Averaging on Manifolds" (2017)</a>, She further develops these themes, viewing singular perturbation problems from a geometric conservation perspective. Published in the Abel Symposium 2016.� </li>�� <li>� <strong>Effective diffusions and intertwined structures.</strong>� In � <a class="addressbg" href="https://arxiv.org/abs/1204.3250">"Effective Diffusions with Intertwined Structures""(2012)</a>, She studied effective behaviour and reduced dynamics for diffusions on manifolds with intertwined structures, allowing the deduction of slowâfast system approximations and their effective motions.� </li>�� <li>� <strong>Averaging principles for integrable stochastic Hamiltonian systems.</strong>� In � <a class="addressbg" href="Averaging.pdf"> "An Averaging Principle for Integrable Stochastic Hamiltonian Systems""</a>, Nonlinearity 21 (2008), pp.803-822, She developed perturbation models for Hamiltonian systems on symplectic manifolds. Built upon completely integrable systems, these models describe approximate Hamiltonian systems with time-dependent and random Hamiltonians, revealing how conserved quantities evolve under random perturbations.� </li>� </ul>�Overall, Xue-Mei Li's work on the multi-scale analysis of classical equations is characterised by its breadth across stoch
1astic analysis, geometric mechanics, and manifold-valued systems, as well as its depth in developing rigorous mathematical frameworks for problems where standard PDE-based or Euclidean averaging techniques fail. Her research has opened pathways for understanding stochastic dynamics in geometric settings with multiple scales, with potential applications in applied mathematics.</li>������ � � <center> <h2 class="title"> Fractional Averaging (non-Markovian multi-scale)</h2> </center>�We study systems of non-Markovian stochastic evolutions with multi-scales in time. � Our aim is to introduce auto-correlated noise instead of white noise in multi-time scale models.�A protype for non-Markovian process with polynomial decay of correlation, stationary increments, �self-similarity, and Gaussianaity is fractional Brownian motion.�Such systems are prevalent. The `derivative' of a fractional Brownian motion is the simplest noise with correlation: �It is Gaussian with stationary increment and $\E(B_{t+s}-B_s)^2=t^{2H}$,�$H\in (0,1)$ is Hurst parameter, BM corresponds to the H=1/2 case. The sample paths of fBM �is Holder with exponent up to H but not including H. The correlation delay of the increments is �t^{2H-2}. WE prove the fractional averaging theorem for an SDE driven by a fractional B�Brownian motion with vector fields depending on a fast evolving stochastic process y_{t/epsilon} �where epsilon is a small parameter. We showed that </p>�(1) For H>1/2, the slow variables converges in probability to that of the solution drive by a fBM �with vector fields simply averaged. Not this differes from the Markovisn case in two ways: � the convergence for the latter is in general in a weak sense and one averages the generator � of the slow variable, not the diffusion vetor fields. ��(2) Such a convergence is false for H<1/2. We show that with a scaling of the order �epsilon^(1/2-H), the convergence holds to that of a Kunita type SDE whose generator is given �by the fractional operator. Such a statement holds also if H>1/2 and the vector fields are centered. �As a consequence we show the convergence of the stochastic flows (the n-point motion) �which appears to be new even in the classical setting H=1/2.���<ul>�<li> Averaging dynamics driven by fractional Brownian motion.� Ann. Probab. (2020), Martin Hairer and Xue-Mei Li� <a class="addressbg" href="https://projecteuclid.org/journals/annals-of-probability/volume-48/issue-4/Averaging-dynamics-driven-by-fractional-Brownian-motion/10.1214/19-AOP1408.full"> article</a>. �Also, https://arxiv.org/abs/1902.11251� </strong></li> ����<li> Generating diffusions with fractional Brownian motion,�To appear: Communications in Mathematical Physics (2021), Martin Hairer and Xue-Mei Li� <a class="addressbg" href="�https://doi.org/10.1007/s00220-022-04462-2"> article</a>. �Also, https://arxiv.org/abs/2109.06948v2� </strong></li> �</ul>�� <center> <h2 class="title"> Homogenisation with non-Markovian and Long Range Dependent Noise </h2> </center>� We begin with proving a Functional limit theorems in the Holder topology for dimension 1 � and apply it for a homogenesation theorems with limit dynamics drivin by fBM' Rosenblatt, � and Hermite processes. To our knowledgem our article is the frist one to give a FCL in the Holder topology.� For higher dimensional cases we proved functional limit theorems in the rough topology allowing� to obtain homogenisation theorems of random ODE's. This is published on the arxiv in the 70 page article� ( arXiv:1911.12600). This article is superceed by late improvements, and therefore not in a journal.� The published article is:� � � <li> Functional limit theorems for the fractional Ornstein-Uhlenbeck process� Journal of Theoretical Probability, Johann Gehringer and Xue-Mei Li� <a class="addressbg" href="�https://link.springer.com/article/10.1007/s10959-020-01044-7"> link to article</a>. �Also, https://arxiv.org/pdf/2006.11540.pdf� </strong></li> � � <li> Diffusive and rough homogenisation in fractional noise field, Johann Gehringer and Xue-Mei Li� <a class="addressbg" href="�https://link.springer.com/article/10.1007/s10959-020-01044-7"> link to article</a>. �Also, https://arxiv.org/pdf/2006.11540.pdf� </strong></li> �� �</ul>�� <center> <h2 class="title"> Homogenisation with weighted average noise </h2>� </center>�We study a general random ODE with fast moving convolution noise. �We treat this as an ODE in a function space�driven by a rough path in the infinite dimensional spaces.�� <li>Functional Limit Theorems for Volterra Processes and Applications to Homogenization, � Johann Gehringer, Xue-Mei Li, and Julian Sieber� <a class="addressbg" href="�https://iopscience.iop.org/article/10.1088/1361-6544/ac4818"> link to article</a>. �Also, https://arxiv.org/abs/2104.06364� </strong></li> � � � <center> <h2 class="title"> Large time behaviour of fractional dynamics </h2>� </center>�By the nature of non-Markovian dynamics, the ergodicity for a stochastic differential equation driven �by a fractional Brownian motion is different. In the Markovian case the invariant measure solves �an elliptic differential equations allowing to study the invariant measure, its tails, kernels, �and smoothness. In our case this is no logner true. Furthermore one cannot prepare the initial �condition to obtain convergence rate estimates. Even more puzzling is that as a �fast dynamics in a two scale system, we appear need to know the large time behaviour of the �dynamics conditioned on its history which could have have a slower rate of convergence. We� prove nevertheless a fractional averaging theorem and obtain Gaussian bounds for � the invariant measure.� � ��� <li>
1On the (Non-)Stationary Density of Fractional-Driven Stochastic Differential Equations, Annals of Probability (To appear) 2023� Xue-Mei Li, Julian Sieber, and Fabien Panloup.� <a class="addressbg" href="�https://arxiv.org/abs/2204.06329"> link to article</a>. � </strong></li> � � <li>Slow-Fast Systems with Fractional Environment and Dynamics� Xue-Mei Li and Julian Sieber. Annals of Applied Probability, 2022, Vol. 32, No. 5, 3964-4003.� <a class="addressbg" href="� https://arxiv.org/abs/2012.01910"> link to article on arxiv</a>;� <a class="addressbg" href="http://dx.doi.org/10.1214/22-AAP1779">on journal site </a>; and� <a class="addressbg" href="https://spiral.imperial.ac.uk/handle/10044/1/93707"> ICL open access site </a>� </strong></li> � � <center> <h2 class="title"> Fractional averaging for stochastic partial differential equations </h2>� </center>� � <li>On the Mild Stochastic Sewing Lemma, SPDE in Random Environment, and Fractional Averaging�by Xue-Mei Li and Julian Sieber is given the best paper award in Stochastics and Dynamics.� <a class="addressbg" href="�https://arxiv.org/pdf/2108.05573.pdf"> link to article</a>. �They began with non-stationary fractional averaging for SPDEs establishing a quantitative stochastic sewing lemma.�� </strong></li> � � � �<center> <h3 class="title"> Fredholm, Hypoelliptic Operators</h3> </center>�Hypoelliptic operators are differential operators that, although possibly degenerate (i.e. not elliptic), still regularise solutions: if the distributional solution is smooth where the data is smooth. In stochastic analysis, hypoelliptic operators often appear as generators of diffusion processes satisfying Hörmanderâs bracket condition, leading to smooth densities and strong probabilistic properties.��In the following article Li has involved Hypoelliptic operators : Homogenisation on Homogeneous Spaces (with an appendix by D. Rumynin), Limits of Random Differential Equations on Manifolds; and Perturbation of Conservation Laws and Averaging on Manifolds.� � � <center> <h3 class="title"> Estimates for Fundamental Solutions of of Parabolic Equations </h3> </center>� <ul>� � This collections of articles focuses on estimates n the kernels of � diffusion processes and inequalities.� � � <li style="text-align: left;">Doubly Damped Parallel translations and Hessian formulas (2017), submitted to the conference on 'Stochastic Partial Differential Equations and Related Fields', Xue-Mei Li.</li>��<Li><a class="addressbg" href="https://www2.warwick.ac.uk/fac/sci/maths/people/staff/xue_mei_li/First-Order-FK.pdf">�First order Feynman-Kac Formula. </A> Xue-Mei Li and James Thompson. 2016, Arxiv�(2016)</Li><a class="addressbg" href="https://arxiv.org/abs/1610.09538"> �<strong>In Stochastic Processes and their applications.�</strong>�<Li> Hessian formulas and estimates for parabolic Schroedinger operators.</a> (2016),� In <strong> J. of Stochastic Analysis </strong>, Vol 2, No. 3 Xue-Mei Li, � <a class="addressbg" href="https://digitalcommons.lsu.edu/cgi/viewcontent.cgi?article=1080&context=josa"> � link to article </a>� https://arxiv.org/abs/1610.09538� � <Li> LOGARITHMIC HEAT KERNEL ESTIMATES WITHOUT CURVATURE RESTRICTIONS.</a> (2021),� To appear in <strong> Annals of Probability </strong>, By XIN CHEN, XUE-MEI LI, and BO WU� <a class="addressbg" href="https://arxiv.org/abs/2106.02746"> � link to article </a>� � <Li> Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in L1.</a> (2021),� To appear in <strong> Potential Analysis </strong>, By N. Gozian, Xue-Mei Li, M. Madiman, C. Roberto,� and P.-M. Samson </strong>� <a class="addressbg" href="https://link.springer.com/content/pdf/10.1007/s11118-021-09934-z.pdf"> � link to article </a>, see also https://arxiv.org/abs/1907.10896.� ��</ul>���� <center> <h3 class="title"> Strictly Local Martingales, Martingales, Semi-martingales</h3> </center>�In the first two articles, the focus is on local martingales which are not true martingales (and name them as strictly local martingales).�In the third, examples of strict local martingales are given.�In the fourth manifold valued martingales are studied �In the fifth, one studies hypoelliptic bridges, i.e. conditioned diffusions associated with hypoelliptic operators. Classical bridges are well-understood for elliptic diffusions (e.g. Brownian bridge), but conditioning hypoelliptic diffusions is subtle due to degeneracies in directions of noise. The main contribution is showing that a class of hypoelliptic bridge processes are semi-martingale and have the L1 integrability property. ��<ul><LI>�<a class="addressbg" href="local-martingales.pdf">The importance of strictly local martingales, �applications to Ornstein-Uhlenbeck processes. </A>
1 K. D. Elworthy, Xue-Mei Li�and M. Yor.� <strong> Probab. Theory and Relat. Fields. </strong>vol 115. p.325-355. (1999).� </LI>� <LI>�<a class="addressbg" href="Strict-local-martingales-Tails.pdf"> On the tails of the supremum and the quadratic variation of strictly�local martingales</A> . K. D. Elworthy, Xue-Mei Li and M. Yor. <strong>Sem. prob.�XXXI, </strong> LNM 1655, June 1997.�</LI>�� � <li style="text-align: left;">Strict Local Martingales: examples. � <strong>, Statistics and Probability Letters</strong> 129(2017) 65-68.� <a href="https://authors.elsevier.com/a/1V9wSc8azys2D"> published version </a>� arxiv:1609.00935 <a href="https://arxiv.org/abs/1609.00935"> Article in Arxiv </a>. Xue-Mei Li.</li>� � <LI>�<a class="addressbg" href="chprob.pdf">Manifold-valued martingales, change of probabilities,�and smoothness of finely harmonic maps. </A> M. Arnaudon, Xue-Mei Li and and A. Thalmaier.� <strong> Annals of Institute Henri Poincare.</strong> Vol. 35, no. 6, p.765-791.�(1999)�</LI>�<Li>� <a class="addressbg" href="http://projecteuclid.org/euclid.ecp/1457617915">�On Hypoelliptic bridge</a> <strong>Electron. Commun. Probab.</strong> 21 (2016) no. 24, 1-12.</li>��</ul>�� <center> <h3 class="title"> Brownian, Generalised and hypoelliptic Bridges</h3> </center>�<ul>�<Li>�`On the Semi-Classical Brownian Bridge Measure' arxiv:1607.06498 <a href="/fac/sci/maths/people/staff/xue_mei_li/IBP-semi-classical-bridge.pdf">Article</a> Electronic Communications in Probability 2017, Vol. 22, paper no. 38, 1-15</li>��<li> Generalised Brownian bridges: examples. arxiv:1612.08716 (2016).� </Li> �</ul>���� <center><h3 class="title"> Existence of Global Smooth Stochastic Flows (strong p-completeness)</h3> </center>�� �<ul> <LI>�<a class="addressbg" href="flow.pdf">Strong p-completeness of stochastic differential�equations and the existence of smooth flows on non-compact manifolds.</A> �<strong> Probab. Theory Relat. Fields.</strong> 100, 4, 485-511. 1994.����<p>�A well-posed SDE is called <strong>complete (or conservative)</strong> if, for every initial point \( x \), its solution \( F_t(x, \omega) \) exists for almost every \( \omega \).�</p>��<p>�An SDE is said to be <strong>strongly complete</strong> (i.e. it has a smooth global solution flow) if there exists a set of full measure in \( \omega \) on which \( F_t(x, \omega) \) is jointly continuous in \( x \) and \( t \). In this case, the regularity of the solution is automatically determined by the regularity of its vector fields. Furthermore, if the adjoint SDE (that is, the SDE with its drift vector field reversed) is also strongly complete, then \( F_t(\cdot, \omega) \) defines diffeomorphisms.�</p>��<p>�When the driving vector fields of an SDE are smooth, solutions are unique. However, uniqueness and existence do not imply that solutions depend continuously on their initial data. In general, the random solution flow must be allowed to start from a random initial condition, for example by restarting from a random variable. The technical difficulty lies in the fact that lifetimes and exit times from relatively compact open sets may not be continuous with respect to initial values. For instance, the lifetime of the ODE�</p>��<p>�\[�\frac{d}{dt} z(t) = z(t)^2�\]��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.�</p>��<p>�On compact smooth manifolds, strong completeness is automatic (this can be proved by lifting the SDE to the diffeomorphism group). For Lipschitz continuous vector fields in Euclidean space, strong completeness can be proved by the fixed point method. However, neither method extends easily to non-compact non-linear manifolds.�</p>��<p>�The question of under what conditions a complete SDE driven by smooth vector fields is strongly complete was solved by Xue-Mei Li. This problem had remained open since the 1970s after the first counterexample was discovered. Their results also yield non-trivial improvements to the known results in \( \mathbb{R}^n \).�</p>��<p>�The strong completeness problem is roughly equivalent to asking whether, for a set of full measure in \( \omega \), a connected relatively compact open set pushed forward by the solution remains connected for all time.�</p>��<p>�The existence of a derivative of \( F_x(\cdot, -) \) in probability is a simpler problem; this leads to solving an SDE known as the derivative flow. Li proved that the order of integration�</p>��<p>�\[�\int f(F_t(x, \omega)) \, dP(\omega),�\]��where \( f \) is a smooth compactly supported function, and differentiation (in probability with respect to the initial point) can be exch
1anged, provided the following holds: the solution starting from a smooth compact curve remains a connected smooth curve. This property is called <strong>strong 1-completeness</strong>.�</p>��<p>�Li also introduced the concept of <strong>strong p-completeness</strong>, and provided examples of SDEs which are strong p-complete but not strongly (p-1)-complete.�</p>��<p>�The main existence theorem, Theorem 4.1, requires control on the moments of the derivative flow. It was found that both the growth of the driving vector fields and the growth of their derivatives are required, and they compensate each other. Applications to SDEs in Euclidean space are given in Section 6, and applications to differentiation of the heat semigroup are in Section 9.�</p>��<p>�This results was extended for non-smooth coefficients, for SDEs in Euclidean space in�<strong>Strong completeness for a class of stochastic differential equations with irregular coefficients</strong>, X. Chen and Xue-Mei Li, <em>Electronic Journal of Probability</em>, 19 (2014), no. 91, 1â34. <a href="http://www.emis.de/journals/EJP-ECP/article/view/3293/2551.html">[article link]</a>, also on arXiv:1402.5079.<br>�The merit of the article is to make clear that the difficulties by the lack of regularity in the driving force and those arising from unbounded derivatives of the vector fields play distinct roles, allowing also the ellipticity of the Markov generator to decay at infinity.�</p>�����<center><h3 class="title"> Complete SDEs Lacking Strong Completeness</h3> </center> � ��� � �<ul> <li><a class="addressbg" href="flow-example.pdf"> Lack of strong completeness for �stochastic flows,</a> �Xue-Mei Li and Michael Scheutzow. <strong> Annals of Probability,</strong> vol 39(4), 1407-1421 (2011).</li> �An example of an SDE with one single real valued Brownian� motion and bounded smooth driving vector fields is shown lacking in strong completeness. In fact the solutions starting from two nearby points, � with the same one dimensional driving noise,� become progressively independent.�</ul>����� <center><h3 class="title">Other Stochastic Flows</h3> </center>� �<ul> � <li style="text-align: left;">Reflected Brownian Motion: selection, approximation, and Linearization. M. Arnaudon and Xue-Mei Li. arxiv:1602.00897 <a href="http://dx.doi.org/10.1214/17-EJP41">Article</a> <strong>Electronic Journal of Probability 2017, </strong>Vol. 22, paper no. 31, 1-55</li>�� They construct a family of approximate stochastic differential equations (SDEs) whose solutions remain within the interior of the manifold and whose damped parallel translations converge to those of the reflected Brownian motion. This reflected Brownian motion solves the heat equation for differential 1-forms under absolute boundary conditions. The construction enables a straightforward transfer of the differentiation formula from manifolds without boundary to those with boundary.��The main difficulty, aside from the inherent nonlinearity, lies in handling the convergence of continuous stochastic processes to limiting processes that may exhibit jumps.��On the real line, the approximation selects the Skorokhod solution rather than the Tanaka solution. The damped parallel transports along the approximating paths, which are sample continuous, converge to a tangent process along the reflected Brownian motion. This limiting process exhibits jumps upon hitting the boundary, occurring precisely at the endpoints of excursions into the interior.���� �<li>�<a class="addressbg" href="barycentre.pdf">Barycentres of probability measures transported by stochastic flows.�</a> M. Arnaudon and Xue-Mei Li. (2005) <strong> Annals of Probability, </strong> vol 33, No.4, 1509-1543.� </li>� �The idea is to transport a subset of a manifold by the solution flow of an SDE. The aim of the article is to study how the center of the mass evolve.�</ul>��<center> <h3 class="title"> Moment Estimates, Moment Stability, Long Time Asymptotics, Non-explosion</h3> </center>��<ul><LI>�<a class="addressbg" href="infinity.pdf">Properties at infinity of diffusion semigroups and�stochastic flows via weak uniform covers. </A> <strong>J. Potential�Anal. </strong> Vol. 3, 339-357. 1994, Xue-Mei Li.�</LI>��<LI>�<a class="addressbg" href="Moment_stability.pdf">
1Stochastic differential equations on non-compact�manifolds: moment stability and its topological consequences, </A> Xue-Mei Li.�<strong> Probab. Theory Relat. Fields</strong>. 100, 4, 417-428. 1994.�</LI>��<li>� <a class="addressbg" href="barycentre-s.pdf">Large time asymptotics of Barycentres of Brownian motions on Hyperbolic spaces �</a> M. Arnaudon and Xue-Mei Li. In Analyse Stochastique et Theorie du Potential. (2003) �</LI>��<LI>�<a class="addressbg" href="exponent.pdf">Derivative flows of stochastic differential equations:�moment exponents and geometric properties.</A> Stochastic Analysis.�In <strong> AMS Proceedings of Symposia in Pure Mathematics</strong>. Eds. Cranston, M. G.�and Pinsky, M. A. p. 565-574. 1995. K. D. Elworthy and Xue-Mei Li.�</LI>�</ul>��� � <center> <h3 class="title"> Interplay between Stochastic Processes and their Underlying Spaces</h3> </center>� <ul>� <LI>�<a class="addressbg" href="group.pdf"> On extensions of Myers' theorem. </A> Xue-Mei Li. <strong>Bull.�London Math. Soc. </strong> 27, 392-396. 1995.�</LI>�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.�<LI>�<a class="addressbg" href="harmonic_forms.pdf"> Bounded and L^2 Harmonic Forms on Universal Covers. �</A> �<strong>Geom. And Funct. Anal.</strong> Vol. 8 , 283-303 (1998). K. D. Elworthy, Xue-Mei Li and and�S. Rosenberg.�</LI>�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.�� <li>�<a class="addressbg" href="Compact.pdf">On compactness of manifolds and existence of certain type �functional inequalities. </a>�Xue-Mei Li and F.-Y. Wang. <strong>Infinite Dimensional Analysis, Quantum Probability and Related Topics.</strong>� Volume 6, pages 29-38, (2003).�</li>� ��<li><a class="addressbg" href="curvature.pdf">Curvature and topology: spectral positivity. </A>�<strong> Methods and Applications of Global Analysis,</strong> 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.�</LI></ul>��� <center> <h3 class="title"> Differential Forms and Harmonic Maps </h3> </center>� �<ul> �<LI>�<a class="addressbg" href="chprob.pdf">Manifold-valued martingales, change of probabilities,�and smoothness of finely harmonic maps. </A> M. Arnaudon, Xue-Mei Li and and A. Thalmaier.�<strong> Annals of Institute Henri Poincare</strong>. Vol. 35, no. 6, p.765-791.�(1999)�</LI>�<LI>�<a class="addressbg" href="Bismut_forms.pdf">Bismut type formula for differential forms.</A>�K. D. Elworthy and Xue-Mei Li. <strong>C. R. Acad. Sci. </strong> t. 327, Série I, 87-92,�(1998)�</LI>�</ul> � �<center> <h3 class="title">The Geometry of Diffusions </h3> </center>� Please also see the two books `The Geometry of Filtering' and `On the Geometry of Diffusion Operators and Stoch
1astic Flows'.�<ul>�<li>�<a class="addressbg" href="principal-1.pdf">�Invariant Diffusions on Principal Bundles. </a>� K. D. Elworthy, Y. LeJan, and Xue-Mei Li. In�<strong>Advanced Studies in Pure Mathematics</strong>,� 31--47, 41, `Stochastic Analysis and Related Topics in Kyoto'.� Math. Soc. Japan, Tokyo, 2004.�</li>����<LI>�<a class="addressbg" href="Tani.pdf">Concerning the geometry of stochastic differential equations�and stochastic flows.</A> In <strong>'New Trends in Stochastic Analysis',�Proc. Taniguchi Symposium, </strong> Sept. 1995, Charingworth, ed. K. D. Elworthy�and S. Kusuoka, I. Shigekawa, World Scientific Press. K. D. Elworthy, Yves Le Jan, Xue-Mei Li�(1997).�</LI>��<li> <a class="addressbg" href="Examples-metric-local.pdf"> Intertwined Diffusions by Examples. </a> Xue-Mei Li.�In <strong>`Stochastic Analysis 2010', </strong> Springer.�ed. D. Crisan. (2010)�</ul>�<ul><center> <h3 class="title"> Books on the Geometry of Diffusion Operators and Semi-elliptic Geometry</h3> </center>� <li> <a class="addressbg" href="http://www2.warwick.ac.uk/fac/sci/maths/people/staff/xue_mei_li/geometry-diffusion-operators-linear-connections.pdf">
1� On the Geometry of Diffusion Operators and Stochastic Flows.</a> <strong>Lecture Notes in Mathematics, </strong> � volume 1720, Springer-Verlag (1999). K. D. Elworthy, Y. LeJan, and Xue-Mei Li.� <a class="addressbg" href="http://www.xuemei.org/Introduction.pdf"> Introduction</a>� </li>� � Here is a **carefully improved and polished version** for clarity, academic precision, and flow:��---��<p>�This is original research, not published elsewhere, focusing on diffusion processes and their associated Riemannian and sub-Riemannian geometry. The main theorem is presented at the beginning of the book (page 8), with the remainder devoted to its applications.�</p>��<p>�The main theorem can be summarised as follows. Consider a sub-elliptic operator of the form:�</p>��<p>�\[ \sum X_i^2 + X_0, \]�</p>��<p>�where \( X_1, \ldots, X_p \) are smooth vector fields. This defines a sub-bundle \( E \) of the tangent bundle. The theorem states that a semi-elliptic second-order differential operator without a zero-order term (i.e. a diffusion operator) in Hörmander form determines a metric linear connection on the sub-bundle with respect to its intrinsic metric. Moreover, every metric connection on such a sub-bundle arises in this way. This connection is characterised by the following property: if \( X \) denotes the bundle map from the trivial bundle over the manifold to the tangent bundle, then the covariant derivative of the vector field \( X(e) \) vanishes in directions where the vector fields are trivial.�</p>��<p>�In general, this connection has torsion. For example, the vector fields induced by an isometric embedding of a Riemannian manifold yield the Levi-Civita connection. The book provides a detailed computation and analysis of this connection: its torsion, curvature, Weitzenböck formulas, and the symmetry properties of its Ricci curvature in relation to torsion. It also studies the associated semigroups. A Hörmander form diffusion operator of constant rank leads to a semi-elliptic stochastic differential equation (SDE), providing probabilistic representations for various problems in analysis, particularly analysis on path spaces. In particular, the work derives Bismut-type formulas, logarithmic Sobolev inequalities, Clark-Ocone formulas, and decompositions of noise.�</p>��<p>�In <em>The Geometry of Filtering</em> (Birkhäuser, 2010), co-authored with K. D. Elworthy and Yves Le Jan, Xue-Mei Li presents original research, not published elsewhere, on intertwined diffusion and differential operators on principal bundles. In this book, she studies second-order differential operators of constant rank, their connections, and associated stochastic flows.�</p>��� <li> <a class="addressbg" �href="http://www2.warwick.ac.uk/fac/sci/maths/people/staff/xue_mei_li/geometry-filter
1ing-xue-mei-li.pdf">� The Geometry of Filtering, </a> <strong> Birkhauser,</strong> (2010). K. D. Elworthy, Yves Le Jan and Xue-Mei Li. </li>��<p>�This book is also original research, studying intertwined diffusion and differential operators on principal bundles. In it, they study second-order differential operators of constant rank, denoted A and B, without zero-order terms, that are intertwined by a map from one manifold to another. The intertwining property determines a splitting of the operator B into the sum of two Hörmander-type operators, referred to as the horizontal and vertical operators. The vertical operator can be viewed as being independent of operator A, leading to applications in stochastic filtering and analysis on path spaces for probability measures determined by degenerate diffusion operators. This method is also useful even when the state spaces of the operators are Euclidean.�</p>���</ul>� <center> <h3 class="title"> Analysis On Path Sapces </h3> </center>��<UL>��<p>�To provide context for the following topics, it is useful to note that one of the most common and topologically non-trivial infinite-dimensional spaces is the space of loops on a Riemannian manifold. The idea is to studying the topology by finite dimensional analogue, establishing De Rham cohomologies. Forcompact smooth Riemannian manifold, the de Rham cohomology coincides with the singular cohomology. While singular cohomology is a topological concept, de Rham cohomology belongs to differential geometry. For non-compact complete Riemannian manifolds, additional considerations are required.�</p>��<p>�The first step towards such a theorem involves an \( L^2 \) Hodge decomposition theorem for differential forms, where the \( L^2 \) space is defined using the volume measure. In the infinite-dimensional loop space, a natural replacement for the volume measure is the Brownian bridge measure. However, loop measures are nowadays replaced in favour of other desired properties in the current study of loop soups.�</p>��<p>�In another direction, Leonard Gross proved the logarithmic Sobolev inequality on the path space (Wiener space) over \( \mathbb{R}^n \), which has become a fundamental tool. There has been substantial effort to extend such inequalities to manifolds, with considerable success on path spaces using the Brownian motion measure. However, on loop spaces, the inequality was shown to hold only for asymptotically Euclidean manifolds. Otherwise, counterexamples arise; for example, the Poincaré inequality fails on the loop space of a compact Riemannian manifold when equipped with the Brownian bridge measure. �</p>��<p>�Developing a Sobolev calculus, including Malliavin derivatives, on path spaces has been necessary, with integration by parts formulas providing a crucial foundation.�</p>��<p>�Specifically, interest focuses on the differential operator \( d^*d \), where the adjoint is taken with respect to the probability distribution of continuous paths of a Brownian motion or Brownian bridge on a finite-dimensional manifold. These inequalities for measures generated by semi-elliptic (degenerate) diffusion operators are discussed in the above mentioned books.�</p>����<center><h3 class="title">Malliavin Calculus over a Manifold</h3></center>��<p>�Li et al. showed that the differentiation formula (BEL formula) is equivalent to the integration by parts formula; see <a class="addressbg" href="IBP-1.pdf">A Class of Integration by Parts Formulae in Stochastic Analysis I</a>. Therefore, it is desirable to obtain a range of such formulas and explore their applications. These results are presented in the following articles. See also <strong>âGeneralised Brownian Bridges: Examplesâ</strong> and <strong>âOn Hypoelliptic Bridgesâ</strong>.�</p>��<ul>� <li>� First Order Feynman-Kac Formula, Xue-Mei Li and J. Thompson. (2016) arXiv:1608.03856.� <a class="addressbg" href="/fac/sci/maths/people/staff/xue_mei_li/First-Order-FK.pdf">Article</a>� </li>�� <li>� <a class="addressbg" href="formulae.pdf">Formulae for the Derivatives of Heat Semigroups</a>, K. D. Elworthy and Xue-Mei Li.� <strong>J. Funct. Anal.</strong> 125(1), 252â286 (1994).� </li>�� <li>� On the Semi-Classical Brownian Bridge Measure, Xue-Mei Li. (2017) arXiv:1607.06498.� <a class="addressbg" href="/fac/sci/maths/people/staff/xue_mei_li/IBP-semi-classical-br
1idge.pdf">Article</a>.� <strong>Electronic Communications in Probability</strong>, Vol. 22, paper no. 38, 1â15.� </li>�� <li>� <a class="addressbg" href="Vilnus.pdf"><strong>Differentiation of Heat Semigroups and Applications</strong></a>, K. D. Elworthy and Xue-Mei Li.� In Probability Theory and Mathematical Statistics (Proc. 6th Vilnius Conference), Eds. B. Grigelionis et al., VSP/TEV, Utrecht and Vilnius, New Titles in Probability and Statistics Series, pp. 239â251 (1994).� </li>�� <li>� <a class="addressbg" href="Degenerate-Diffusion.pdf">Integration by Parts Formulae for Degenerate Diffusion Measures on Path Spaces and Diffeomorphism Groups</a>, K. D. Elworthy, Yves Le Jan, and Xue-Mei Li.� <strong>C. R. Acad. Sci.</strong> t.323, série 1, pp. 921â926 (1997).� </li>�</ul>��<center><h3 class="title">Generalised KPP Equation</h3></center>��<ul>� <li>� <a class="addressbg" href="reaction_diffusion.pdf">Gradient Estimates and the Smooth Convergence of Approximate Travelling Waves for Reaction-Diffusion Equations</a>, Xue-Mei Li and H. Z. Zhao.� <strong>Nonlinearity</strong>, 9, 459â477 (1996).� </li>�</ul>��<p>�They discuss semilinear parabolic equations (reaction-diffusion equations) on \\( \\mathbb{R}^n \\) with a small parameter. It is well known that, under suitable conditions, there exists a function \\( V \\) such that solutions converge to 0 or 1 depending on whether \\( V(t,x) \\) is negative or positive. They study the convergence of the first spatial derivatives of the solutions and prove that these derivatives also converge on the trough, where \\( V(t,x) < 0 \\).�</p>��<center><h3 class="title">Spectral Gap and Poincaré Inequality on Path Spaces</h3></center>��<p>�The focus is on the Brownian bridge measure on the space of continuous loops over finite-dimensional manifolds.�</p>��<ul>� <li><a class="addressbg" href="Concrete-Estimates.pdf">A Concrete Estimate for the Weak Poincaré Inequality on Loop Space</a>, X. Chen, Xue-Mei Li, and B. Wu. <strong>Probability Theory and Related Fields</strong> (2011) 151:559â590.</li>�� <li><a class="addressbg" href="Spectral-Hyperbolic.pdf">A Spectral Gap for the Brownian Bridge Measure on Hyperbolic Spaces</a>, X. Chen, Xue-Mei Li, and B. Wu. In <strong>Progress in Analysis and its Applications</strong>, pp. 398â404, World Scientific Publishing, Hackensack, NJ, 2010.</li>�� <li><a class="addressbg" href="http://www.xuemei.org/PoincareInequalityLoopSpaces.pdf">A Poincaré Inequality on Loop Spaces</a>, X. Chen, Xue-Mei Li, and B. Wu. <strong>Journal of Functional Analysis</strong>, vol. 259(6), pp. 1421â1442 (2010).</li>�</ul>��<center><h3 class="title">Toward an L² Hodgeâde Rham Theory on Infinite-Dimensional Space
1s</h3></center>��<p>�The well-known de Rham theorem relates de Rham cohomology, a differential geometric concept, with singular cohomology, a topological concept, yielding far-reaching consequences. As a step towards such a theory on infinite-dimensional spaces, the existence of an L² Hodge decomposition theorem is studied on path spaces, which are manifolds modelled on infinite-dimensional Banach spaces. A key difficulty is the lack of natural L² tensor spaces for the tangent spaces.�</p>��<ul>� <li><a class="addressbg" href="ICMart.pdf">Geometric Stochastic Analysis on Path Spaces</a>, K. D. Elworthy and Xue-Mei Li. <strong>Proceedings of the International Congress of Mathematicians</strong>, pp. 575â594 (2006).</li>�� <li><a class="addressbg" href="L2-theory-1.pdf">An L² Theory for Differential Forms on Path Spaces I</a>, K. D. Elworthy and Xue-Mei Li. <strong>Journal of Functional Analysis</strong>, 254 (2008), pp. 196â245.</li>�� <li><a class="addressbg" href="vector-fields.pdf">Some Families of q-Vector Fields on Path Spaces</a>, K. D. Elworthy and Xue-Mei Li. <strong>Infinite Dimensional Analysis, Quantum Probability and Related Topics</strong>, 6, 1â27 (2003).</li>�� <li><a class="addressbg" href="Short-Hodge-2.pdf">Hodgeâde Rham Decomposition for an L² Space of Differential 2-Forms on Path Spaces</a>, K. D. Elworthy and Xue-Mei Li (2002).</li>�� <li><a class="addressbg" href="Hodge-1.pdf">Special Itô Maps and an L² Hodge Theory for One Forms on Path Spaces</a>, K. D. Elworthy and Xue-Mei Li. In <strong>Stochastic Processes, Physics and Geometry: New Interplays</strong>, I (Leipzig, 1999), pp. 145â162, CMS Conference Proceedings, vol. 28, American Mathematical Society, Providence, RI, 2000.</li>�</ul>��<center><h3 class="title">Toward a Sobolev Calculus on Path and Loop Spaces</h3></center>��<p>�The aim is to establish a Sobolev calculus in infinite-dimensional spaces, where derivatives are understood in the sense of Malliavin calculus. The idea is to construct charts that pull the corresponding calculus from Wiener space. One technique involves using stochastic flows from SDEs whose measures provide the basis for the analysis and whose vector fields induce a suitable covariant derivative on the finite-dimensional manifold, combined with stochastic filtering techniques to keep all concepts intrinsic.�</p>��<ul>� <li><a class="addressbg" href="Ito-map.pdf">Itô Maps and Analysis on Path Spaces</a>, K. D. Elworthy and Xue-Mei Li. <strong>Mathematische Zeitschrift</strong> (2007) 257:643â706.</li>�� <li><a class="addressbg" href="SDE-paths.pdf">The Stochastic Differential Equation Approach to Analysis on Path Space</a>, in <strong>New Trends in Stochastic Analysis and Related Topics</strong>, pp. 207â226 (2012).</li>�� <li><a class="addressbg" href="Markov.uniqueness.pdf">Intertwining and the Markov Uniqueness Problem on Path Spaces</a>, K. D. Elworthy and Xue-Mei Li. In <strong>Stochastic Partial Differential Equations and Applications VII</strong>, eds. G. Da Prato and L. Tubaro (2005).</li>�� <li>GrossâSobolev Spaces on Path Manifolds: Uniqueness and Intertwining by Itô Maps, K. D. Elworthy and Xue-Mei Li. <strong>C. R. Acad. Sci. Paris</strong>, Ser. I 337 (2003), 741â744.</li>�</ul>��<p>�Here, they attempt to prove Markov uniqueness for the operator d*d. They manage to show that the closure of BC² functions coincides with that of smooth compactly supported functions. However, the gap remains in proving the same for BC¹ functions.�</p>��</BODY>�</HTML>�
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