Xue-mei Li | Imperial College London (original) (raw)

Papers by Xue-mei Li

Journal of the Mathematical Society of Japan, 2018

The version presented here may differ from the published version or, version of record, if you wi... more The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP URL' above for details on accessing the published version and note that access may require a subscription.

$Research paper thumbnail of Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">L</mi><mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb {L}^{1}</annotation></semantics></math>L1$

Potential Analysis, 2021

It is well known that some important Markov semi-groups have a "regularization effect"-as for exa... more It is well known that some important Markov semi-groups have a "regularization effect"-as for example the hypercontractivity property of the noise operator on the Boolean hypercube or the Ornstein-Uhlenbeck semi-group on the real line, which applies to functions in L p for p > 1. Talagrand had conjectured in 1989 that the noise operator on the Boolean hypercube has a further subtle regularization property for functions that are just integrable, but this conjecture remains open. Nonetheless, the Gaussian analogue of this conjecture was proven in recent years by Eldan-Lee and Lehec, by combining an inequality for the log-Hessian of the Ornstein-Uhlenbeck semi-group with a new deviation inequality for log-semi-convex functions under Gaussian measure. In this work, we explore the question of how much more general this phenomenon is. Specifically, our first goal is to explore the validity of both these ingredients for some diffusion semi-groups in R n , as well as for the M/M/∞ queue on the non-negative integers and the Laguerre semi-groups on the positive real line. Our second goal is to prove a one-dimensional regularization effect for these settings, even in those cases where these ingredients are not valid.

Computation and Combinatorics in Dynamics, Stochastics and Control, 2018

We prove a stochastic averaging theorem for stochastic differential equations in which the slow a... more We prove a stochastic averaging theorem for stochastic differential equations in which the slow and the fast variables interact. The approximate Markov fast motion is a family of Markov process with generator L x for which we obtain a locally uniform law of large numbers and obtain the continuous dependence of their invariant measures on the parameter x. These results are obtained under the assumption that L x satisfies Hörmander's bracket conditions, or more generally L x is a family of Fredholm operators with sub-elliptic estimates. On the other hand a conservation law of a dynamical system can be used as a tool for separating the scales in singular perturbation problems. We also study a number of motivating examples from mathematical physics and from geometry where we use non-linear conservation laws to deduce slow-fast systems of stochastic differential equations.

Electronic Communications in Probability, 2016

A conditioned hypoelliptic process on a compact manifold, satisfying the strong Hörmander's condi... more A conditioned hypoelliptic process on a compact manifold, satisfying the strong Hörmander's condition, is a hypoelliptic bridge. If the Markov generator satisfies the two step strong Hörmander condition, the drift of the conditioned hypoelliptic bridge is integrable on [0, 1] and the hypoelliptic bridge is a continuous semi-martingale.

Probability Theory and Related Fields, 1994

Here we discuss the regularity of solutions of SDE's and obtain conditions under which a SDE on a... more Here we discuss the regularity of solutions of SDE's and obtain conditions under which a SDE on a complete Riemannian manifold M has a global smooth solution flow, in particular improving the usual global Lipschitz hypothesis when M-R n. There are also results on non-explosion of diffusions.

arXiv (Cornell University), Oct 13, 2008

If L is a second order differential operator on a manifold M, denote by σ L : T * M → T M its sym... more If L is a second order differential operator on a manifold M, denote by σ L : T * M → T M its symbol determined by df σ L (dg) = 1 2 L (f g) − 1 2 (Lf)g − 1 2 f (Lg), for C 2 functions f, g. We will often write σ L (ℓ 1 , ℓ) for ℓ 1 σ L (ℓ 2) and consider σ L as a bilinear form on T * M. Note that it is symmetric. The operator is said to be semi-elliptic if σ L (ℓ 1 , ℓ 2) 0 for all ℓ 1 , ℓ 2 ∈ T u M * , all u ∈ M, and elliptic if the inequality holds strictly. Ellipticity is equivalent to σ L being onto. Definition 1.0.1 A semi-elliptic smooth second order differential operator L is said to be a diffusion operator if L1 = 0.

For a compact Riemannian manifold the space L 2 A of L 2 differential forms decomposes into the d... more For a compact Riemannian manifold the space L 2 A of L 2 differential forms decomposes into the direct sum of three spaces, the Hodge decomposition,

Lecture Notes in Pure and Applied Mathematics, 2005

Techniques of intertwining by Itô maps are applied to uniqueness questions for the Gross-Sobolev ... more Techniques of intertwining by Itô maps are applied to uniqueness questions for the Gross-Sobolev derivatives that arise in Malliavin calculus on path spaces. In particular claims in our article [Elworthy-Li3] are corrected and put in the context of the Markov uniqueness problem and weak differentiability. Full proofs in greater generality will appear in [Elworthy-Li2].

Elliptic stochastic differential equations (SDE) make sense when the coefficients are only contin... more Elliptic stochastic differential equations (SDE) make sense when the coefficients are only continuous. We study the corresponding linearized SDE whose coefficients are not assumed to be locally bounded. This leads to existence of W 1,p loc solution flows for elliptic SDEs with Hölder continuous and ∩ p W 1,p loc coefficients. Furthermore an approximation scheme is studied from which we obtain a representation for the derivative of the Markov semigroup, and an integration by parts formula.

Journal of Functional Analysis, 2010

This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright

Geometric And Functional Analysis, 1998

… , physics, and geometry: new interplays: a …, 2000

Cornell University - arXiv, Nov 19, 2019

Some families of H-valued vector fields with calculable Lie brackets are given. These provide exa... more Some families of H-valued vector fields with calculable Lie brackets are given. These provide examples of vector fields on path spaces with a divergence and we show that versions of Bismut type formulae for forms on a compact Riemannian manifold arise as projections of the infinite dimensional theory.

This article was published in an Elsevier journal. The attached copy is furnished to the author f... more This article was published in an Elsevier journal. The attached copy is furnished to the author for non-commercial research and education use, including for instruction at the author’s institution, sharing with colleagues and providing to institution administration. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit:

$Research paper thumbnail of Diffusive and rough homogenisation in fractional noise field$

arXiv: Probability, 2020

With recently developed tools, we prove a homogenisation theorem for a random ODE with short and ... more With recently developed tools, we prove a homogenisation theorem for a random ODE with short and long-range dependent fractional noise. The effective dynamics are not necessarily diffusions, they are given by stochastic differential equations driven simultaneously by stochastic processes from both the Gaussian and the non-Gaussian self-similarity universality classes. A key lemma for this is the `lifted' joint functional central and non-central limit theorem in the rough path topology.

The main results of the article are short time estimates and asymptotic estimates for the first t... more The main results of the article are short time estimates and asymptotic estimates for the first two order derivatives of the logarithmic heat kernel of a complete Riemannian manifold. We remove all curvature restrictions and also develop several techniques. A basic tool developed here is intrinsic stochastic variations with prescribed second order covariant differentials, allowing to obtain a path integration representation for the second order derivatives of the heat semigroup Pt on a complete Riemannian manifold, again without any assumptions on the curvature. The novelty is the introduction of an 2 term in the variation allowing greater control. We also construct a family of cut-off stochastic processes adapted to an exhaustion by compact subsets with smooth boundaries, each process is constructed path by path and differentiable in time, furthermore the differentials have locally uniformly bounded moments with respect to the Brownian motion measures, allowing to by-pass the lack ...

Journal of Theoretical Probability, 2020

We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein–Uhle... more We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein–Uhlenbeck process, providing the foundation for the fluctuation theory of slow/fast systems driven by both long- and short-range-dependent noise. The limit process has both Gaussian and non-Gaussian components. The theorem holds for any$$L^2$$L2functions, whereas for functions with stronger integrability properties the convergence is shown to hold in the Hölder topology, the rough topology for processes in$$C^{\frac{1}{2}+}$$C12+. This leads to a ‘rough creation’ / ‘rough homogenization’ theorem, by which we mean the weak convergence of a family of random smooth curves to a non-Markovian random process with non-differentiable sample paths. In particular, we obtain effective dynamics for the second-order problem and for the kinetic fractional Brownian motion model.

Electronic Journal of Probability, 2017

We construct a family of SDEs with smooth coefficients whose solutions select a reflected Brownia... more We construct a family of SDEs with smooth coefficients whose solutions select a reflected Brownian flow as well as a corresponding stochastic damped transport process (Wt), the limiting pair gives a probabilistic representation for solutions of the heat equations on differential 1-forms with the absolute boundary conditions. The transport process evolves pathwise by the Ricci curvature in the interior, by the shape operator on the boundary where it is driven by the boundary local time, and with its normal part erased at the end of the excursions to the boundary of the reflected Brownian motion. On the half line, this construction selects the Skorohod solution (and its derivative with respect to initial points), not the Tanaka solution; on the half space it agrees with the construction of N. Ikeda and S. Watanabe [29] by Poisson point processes. The construction leads also to an approximation for the boundary local time, in the topology of uniform convergence but not in the semi-martingale topology, indicating the difficulty in proving convergence of solutions of a family of random ODE's to the solution of a stochastic equation driven by the local time and with jumps. In addition, we obtain a differentiation formula for the heat semi-group with Neumann boundary condition and prove also that (Wt) is the weak derivative of a family of reflected Brownian motions with respect to the initial point.

Frontiers in Mathematics, 2010

Let π : P → M be a smooth principal bundle with structure group G. This means that there is a C ∞... more Let π : P → M be a smooth principal bundle with structure group G. This means that there is a C ∞ right multiplication P × G → P , u → u • g say, of the Lie group G such that π identifies the space of orbits of G with the manifold M and π is locally trivial in the sense that each point of M has an open neighbourhood U with a diffeomorphism