Imperial College London

Professor Xue-Mei Li

Faculty of Natural SciencesDepartment of Mathematics

Chair in Probability and Stochastic Analysis
 
 
 
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Contact

 

+44 (0)20 7594 3709xue-mei.li Website CV

 
 
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Location

 

6M51Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Publication Type
Year
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74 results found

Li X-M, Sieber J, 2022, Slow-fast systems with fractional environment and dynamics, Annals of Applied Probability, Vol: 32, Pages: 3964-4003, ISSN: 1050-5164

We prove a fractional averaging principle for interacting slow-fast systems. The mode of convergence is in Hölder norm in probability. The main technical result is a quenched ergodic theorem on the conditioned fractional dynamics. We also establish geometric ergodicity for a class of fractional-driven stochastic differential equations, improving a recent result of Panloup and Richard.MSC2010: 60G22, 60H10, 37A25.Keywords: Fractional Brownian motion, averaging, slow-fast system, quenched ergodic theorem, rate of convergence to equilibrium.

Journal article

Li X-M, Chen X, Wu B, 2022, LOGARITHMIC HEAT KERNEL ESTIMATES WITHOUT CURVATURE RESTRICTIONS, Annals of Probability, ISSN: 0091-1798

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 pre- scribed second order covariant differentials, allowing to obtain a path inte- gration 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 ε2 term in the variation allowing greater control. We also construct a family of cut-off stochastic pro- cesses adapted to an exhaustion by compact subsets with smooth boundaries, each process is constructed path by path and differentiable in time, further- more the differentials have locally uniformly bounded moments with respect to the Brownian motion measures, allowing to by-pass the lack of continuity of the exit time of the Brownian motions on its initial position.

Journal article

Hairer M, Li X-M, 2022, Generating diffusions with fractional Brownian motion, Communications in Mathematical Physics

We study fast / slow systems driven by a fractional Brownian motion $B$ withHurst parameter $H\in (\frac 13, 1]$. Surprisingly, the slow dynamic convergeson suitable timescales to a limiting Markov process and we describe itsgenerator. More precisely, if $Y^\varepsilon$ denotes a Markov process withsufficiently good mixing properties evolving on a fast timescale $\varepsilon\ll 1$, the solutions of the equation $$ dX^\varepsilon = \varepsilon^{\frac12-H} F(X^\varepsilon,Y^\varepsilon)\,dB+F_0(X^\varepsilon,Y^\varepsilon)\,dt\;$$ converge to a regular diffusion without having to assume that $F$ averagesto $0$, provided that $H< \frac 12$. For $H > \frac 12$, a similar resultholds, but this time it does require $F$ to average to $0$. We also prove thatthe $n$-point motions converge to those of a Kunita type SDE. One nice interpretation of this result is that it provides a continuousinterpolation between the homogenisation theorem for random ODEs with rapidlyoscillating right-hand sides ($H=1$) and the averaging of diffusion processes($H= \frac 12$).

Journal article

Li X-M, Sieber J, 2022, Mild Stochastic Sewing Lemma, SPDE in Random Environment, and Fractional Averaging, Stochastics and Dynamics, ISSN: 0219-4937

Our first result is a stochastic sewing lemma with quantitative estimates formild incremental processes, with which we study SPDEs driven by fractionalBrownian motions in a random environment. We obtain uniform $L^p$-bounds. Oursecond result is a fractional averaging principle admitting non-stationary fastenvironments. As an application, we prove a fractional averaging principle forSPDEs.

Journal article

Gehringer J, Li X-M, 2022, Functional limit theorems for the fractional Ornstein-Uhlenbeck process, Journal of Theoretical Probability, Vol: 35, Pages: 426-456, ISSN: 0894-9840

We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein-Uhlenbeckprocess, providing the foundation for the fluctuation theory of slow/fast systems driven by both long and shortrange dependent noise. The limit process has both Gaussian and non-Gaussian components. The theoremholds for any L2functions, whereas for functions with stronger integrability properties the convergence isshown to hold in the Hölder topology, the rough topology for processes in C12 +. This leads to a ‘roughcreation’ / ‘rough homogenization’ theorem, by which we mean the weak convergence of a family of randomsmooth curves to a non-Markovian random process with non-differentiable sample paths. In particular, weobtain effective dynamics for the second order problem and for the kinetic fractional Brownian motion model.

Journal article

Gehringer J, Li X-M, Sieber J, 2022, Functional limit theorems for volterra processes and applications to homogenization, Nonlinearity, Vol: 35, Pages: 1-37, ISSN: 0951-7715

We prove an enhanced limit theorem for additive functionals of a multi-dimensional Volterra process (yt)t≥0 in the rough path topology. As an application, we establish weak convergence as ε→0 of the solution of the random ordinary differential equation (ODE) ddtxεt=1ε√f(xεt,ytε) and show that its limit solves a rough differential equation driven by a Gaussian field with a drift coming from the Lévy area correction of the limiting rough driver. Furthermore, we prove that the stochastic flows of the random ODE converge to those of the Kunita type Itô SDE dxt=G(xt,dt), where G(x,t) is a semi-martingale with spatial parameters.

Journal article

Li X-M, Hairer M, 2021, Generating diffusions with fractional Brownian motion, Publisher: ArXiv

We study fast / slow systems driven by a fractional Brownian motion B with Hurst parameter H∈(13,1]. Surprisingly, the slow dynamic converges on suitable timescales to a limiting Markov process and we describe its generator. More precisely, if Yε denotes a Markov process with sufficiently good mixing properties evolving on a fast timescale ε≪1, the solutions of the equationdXε=ε12−HF(Xε,Yε)dB+F0(Xε,Yε)dtconverge to a regular diffusion without having to assume that F averages to 0, provided that H<12. For H>12, a similar result holds, but this time it does require F to average to 0. We also prove that the n-point motions converge to those of a Kunita type SDE. One nice interpretation of this result is that it provides a continuous interpolation between the homogenisation theorem for random ODEs with rapidly oscillating right-hand sides (H=1) and the averaging of diffusion processes (H=12).

Working paper

Li X-M, Sieber J, 2021, Mild Stochastic Sewing Lemma, SPDE in Random Environment, and Fractional Averaging

Our first result is a stochastic sewing lemma with quantitative estimates for mild incremental processes, with which we study SPDEs driven by fractional Brownian motions in a random environment. We obtain uniform Lp-bounds. Our second result is a fractional averaging principle admitting non-stationary fast environments. As an application, we prove a fractional averaging principle for SPDEs.

Working paper

Li X-M, Gehringer J, 2021, Functional limit theorems for the fractional Ornstein-Uhlenbeck process, Publisher: ArXiv

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 such a noise. Our main contribution is on the joint convergence to a limit with both Gaussian and non-Gaussian components. This is valid for any L2 functions, whereas for functions with stronger integrability properties the convergence is shown to hold in the Hölder topology. As an application we prove a `rough creation' result, i.e. the weak convergence of a family of random smooth curves to a non-Markovian random process with rough sample paths. This includes the second order problem and the kinetic fractional Brownian motion model.

Working paper

Li X-M, 2021, Hessian formulas and estimates for parabolic Schrödinger operators, Journal of Stochastic Analysis, Vol: 2, Pages: 1-54, ISSN: 2689-6931

We study the Cauchy problem for the parabolic equation ∂∂t = L and theh-Brownian motion which is the Markov process with the weighted Laplacian 12 ∆h :=12 ∆ +∇h where ∆ the Laplace-Beltrami operator on M, and h, V real valued functionson M and L is the weighted Schrodinger operator ¨ L = 12 ∆ + ∇h − V .We first obtain new geometric criteria for the gradient stochastic differential equation(SDE) with generator 12 ∆h: non-explosion, strong 1-completeness, moment bounds,and exponential integrability. We then study the linearisation problem associated withthe gradient SDE, introduce also a doubly damped stochastic parallel transport on tensors, involving only geometric quantities. Together with the stochastic damped transport this allows to obtain a new Hessian formula for the weighted heat semi-group,obtained with a hybrid formula Hess(Phtf)(v2, v1) = E [∇df(Wt(v2), Wt(v1))] +Ehdf(W(2)t(v1, v2))i, and a corresponding formula for etL. These formulae are thenused for obtaining path integration formula for Hess Ph,Vtf(v1, v2), a 2nd order Feynman - Kac formula, based on path integration, not involving any derivatives of f or V.With these intrinsic second order Feynman-Kac formula, global estimates are obtained for these semi-groups, their derivatives, and that of their fundamental solutions areobtained. These estimates are in terms of bounds on Ric −2 Hess h, on the curvatureoperator, and on the cyclic sum of the gradient of the Ricci tensor.Finally, for manifolds with a pole, we prove that the Hessian of the fundamentalsolution is the product of an exact Gaussian term with a term involving the semi-classicalbridge, the latter is further estimated to lead to Hessian estimates. Precise estimates arethen obtained for the derivatives of the logarithmic heat kernels.

Journal article

Gozlan N, Li X-M, Madiman M, Roberto C, Samson P-Met al., 2021, Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in L1, Potential Analysis, ISSN: 0926-2601

In 1989, Talagrand proposed a conjecture regarding the regularization effecton integrable functions of a natural Markov semigroup on the Boolean hypercube.While this conjecture remains unresolved, the analogous conjecture for theOrnstein-Uhlenbeck semigroup was recently resolved by Eldan-Lee and Lehec, bycombining an inequality for the log-Hessian of this semigroup with a newdeviation inequality for log-semiconvex functions under Gaussian measure. Ourfirst goal is to explore the validity of both these ingredients for somediffusion semigroups in R n as well as for the M/M/$\infty$ queue on thenon-negative integers. Our second goal is to prove an analogue of Talagrand'sconjecture for these settings, even in those cases where these ingredients arenot valid.

Journal article

Gehringer J, Li X-M, Sieber J, 2021, Functional limit theorems for volterra processes and applications tohomogenization, Publisher: arXiv

We prove an enhanced limit theorem for additive functionals of amultidimensional Volterra process $(y_t)_{t\geq 0}$. As an application, weestablish weak convergence of the solutions of rough differential equations(RDE) of the form $$ dx^\varepsilon_t=\frac 1 {\sqrt \varepsilon}f(x_t^\varepsilon,y_{\frac{t}{\varepsilon}})\,dt+g(x_t^\varepsilon)\,d\mathbf{B}_t,$$and identify their limits as solutions of an RDE driven by a Gaussian fieldwith a drift coming from the L\'evy area correction of the limiting roughdriver. The equation models a passive tracer in a random field. In particular if $h$ is random field such that $h(x, \cdot)$ asemi-martingale with spatial parameter $x$, we show that the solutions of theequations $$ dx^\varepsilon_t=\frac 1 {\sqrt \epsilon}f(x_t^\varepsilon,y_{\frac{t}{\varepsilon}})\,dt+h(x_t^\varepsilon, dt),$$ converge weakly to that of a Kunita type It\^o SDE $dx_t=G(x_t,dt)$ where$G(x,t)$ is a semi-martingale with spatial parameters. Furthermore the$N$-point motions converge.

Working paper

Li X-M, Sieber J, 2020, Slow-fast systems with fractional environment and dynamics, Publisher: arXiv

We prove an averaging principle for interacting slow-fast systems driven byindependent fractional Brownian motions. The mode of convergence is in H\"oldernorm in probability. We also establish geometric ergodicity for a class offractional-driven stochastic differential equations, partially improving arecent result of Panloup and Richard.

Working paper

Gehringer J, Li X-M, 2020, Rough homogenisation with fractional dynamics, Publisher: arXiv

We review recent developments of slow/fast stochastic differential equations,and also present a new result on Diffusion Homogenisation Theory with fractional and non-strong-mixing noiseand providing new examples. The emphasise of the review will be on the recently developed effectivedynamic theory for two scale random systems with fractional noise: StochasticAveraging and `Rough Diffusion Homogenisation Theory'. We also study thegeometric models with perturbations to symmetries.

Working paper

Gehringer J, Li X-M, 2020, Diffusive and rough homogenisation in fractional noise field, Publisher: arXiv

With recently developed tools, we prove a homogenisation theorem for a randomODE with short and long-range dependent fractional noise. The effectivedynamics are not necessarily diffusions, they are given by stochasticdifferential equations driven simultaneously by stochastic processes from boththe Gaussian and the non-Gaussian self-similarity universality classes. A keylemma for this is the `lifted' joint functional central and non-central limittheorem in the rough path topology.

Working paper

Li X-M, Gehringer J, 2019, Homogenization with fractional random fields, Publisher: arXiv

We consider a system of differential equations in a fast long range dependent random environment and prove a homogenization theorem involving multiple scaling constants. The effective dynamics solves a rough differential equation, which is `equivalent' to a stochastic equation driven by mixed Itô integrals and Young integrals with respect to Wiener processes and Hermite processes. Lacking other tools we use the rough path theory for proving the convergence, our main technical endeavour is on obtaining an enhanced scaling limit theorem for path integrals (Functional CLT and non-CLT's) in a strong topology, the rough path topology, which is given by a Hölder distance for stochastic processes and their lifts. In dimension one we also include the negatively correlated case, for the second order / kinetic fractional BM model we also bound the error.

Working paper

Hairer M, Li X-M, 2019, Averaging dynamics driven by fractional Brownian motion, Annals of Probability, Vol: 48, Pages: 1826-1860, ISSN: 0091-1798

We consider slow / fast systems where the slow system is driven by fractionalBrownian motion with Hurst parameter $H>{1\over 2}$. We show that unlike in thecase $H={1\over 2}$, convergence to the averaged solution takes place inprobability and the limiting process solves the 'na\"ively' averaged equation.Our proof strongly relies on the recently obtained stochastic sewing lemma.

Journal article

Li X-M, 2018, Perturbation of conservation laws and averaging on manifolds, Computation and Combinatorics in Dynamics, Stochastics and Control, Editors: Celledoni, Di Nunno, Ebrahimi-Fard, Munthe-Kaas, Publisher: Springer, Pages: 499-550, ISBN: 978-3-030-01592-3

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 Lx for which we obtain a quantitative 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 Lx satisfies Hörmander’s bracket conditions, or more generally Lx is a family of Fredholm operators with sub-elliptic estimates. For stochastic systems in which the slow and the fast variable are not separate, conservation laws are essential ingredients for separating the scales in singular perturbation problems we demonstrate this by a number of motivating examples, from mathematical physics and from geometry, where conservation laws taking values in non-linear spaces are used to deduce slow-fast systems of stochastic differential equations.

Book chapter

Li X-M, Thompson J, 2018, First order Feynman-Kac formula, Stochastic Processes and their Applications, Vol: 128, Pages: 3006-3029, ISSN: 0304-4149

We study the parabolic integral kernel associated with the weighted Laplacianand the Feynman-Kac kernels. For manifold with a pole we deduce formulas andestimates for them and for their derivatives, given in terms of a Gaussian termand the semi-classical bridge. Assumptions are on the Riemannian data.

Journal article

Li X-M, 2018, Doubly damped stochastic parallel translations and Hessian formulas, International Conference on Stochastic Partial Differential Equations and Related Fields, Publisher: Springer

We study the Hessian of the solutions of time-independent Schrödinger equations, aiming to obtain as large a class as possible of complete Riemannian manifolds for which the estimate C(1t+d2t2) holds. For this purpose we introduce the doubly damped stochastic parallel transport equation, study them and make exponential estimates on them, deduce a second order Feynman-Kac formula and obtain the desired estimates. Our aim here is to explain the intuition, the basic techniques, and the formulas which might be useful in other studies

Conference paper

Hairer X, 2018, Generalised Brownian bridges: examples, Markov Processes and Related Fields, Vol: 24, Pages: 151-163, ISSN: 1024-2953

We observe that the probability distribution of the Brownian motion with drift −cx/(1−t) where c≠1 is singular with respect to that of the classical Brownian bridge measure on [0,1], while their Cameron\tire Martin spaces are equal set-wise if and only if c>1/2, providing also examples of exponential martingales on [0,1) not extendable to a continuous martingale on [0,1]. Other examples of generalised Brownian bridges are also studied.

Journal article

Li X-M, 2018, Doubly Damped Stochastic Parallel Translations and Hessian Formulas, Conference In Honor of Michael Röckner, SPDERF

We study the Hessian of the solutions of time-independent Schrodinger ¨equations, aiming to obtain as large a class as possible of complete Riemannianmanifolds for which the estimateC(1t +d2t2) holds. For this purpose we introduce thedoubly damped stochastic parallel transport equation, study them and make exponentialestimates on them, deduce a second order Feynman-Kac formula and obtainthe desired estimates. Our aim here is to explain the intuition, the basic techniques,and the formulas which might be useful in other studies.

Conference paper

Li X-M, 2018, Homogenisation on homogeneous spaces, Journal of Mathematical Society of Japan, Vol: 70, Pages: 519-572, ISSN: 0025-5645

Motivated by collapsing of Riemannian manifolds and inhomogeneous scaling of left invariant Riemannian metrics on a real Lie group $G$ with a sub-group $H$, we introduce a family of interpolation equations on $G$ with a parameter $\epsilon&gt;0$, interpolating hypo-elliptic diffusions on $H$ and translates of exponential maps on $G$ and examine the dynamics as $\epsilon\to 0$. When $H$ is compact, we use the reductive homogeneous structure of Nomizu to extract a converging family of stochastic processes (converging on the time scale $1/\epsilon$), proving the convergence of the stochastic dynamics on the orbit spaces $G/H$ and their parallel translations, providing also an estimate on the rate of the convergence in the Wasserstein distance. Their limits are not necessarily Brownian motions and are classified algebraically by a Peter-Weyl's theorem for real Lie groups and geometrically using a weak notion of the naturally reductive property; the classifications allow to conclude the Markov property of the limit process. This can be considered as "taking the adiabatic limit" of the differential operators $\mathcal{L}^\epsilon=(1/\epsilon) \sum_k (A_k)^2+(1/\epsilon) A_0+Y_0$ where $Y_0, A_k$ are left invariant vector fields and $\{A_k\}$ generate the Lie-algebra of $H$.

Journal article

Arnaudon M, Li X-M, 2017, Reflected Brownian motion: selection, approximation and linearization, Electronic Journal of Probability, Vol: 22, Pages: 1-55, ISSN: 1083-6489

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)(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)(Wt) is the weak derivative of a family of reflected Brownian motions with respect to the initial point.

Journal article

Li X-M, 2017, Strict local martingales: Examples, STATISTICS & PROBABILITY LETTERS, Vol: 129, Pages: 65-68, ISSN: 0167-7152

We show that a continuous local martingale is a strict local martingale if its supremum process is not in Lα for a positive number α smaller than 1. Using this we construct a family of strict local martingales.

Journal article

Li X-M, 2017, On the Semi-classical Brownian Bridge Measure, Electronic Communications in Probability, Vol: 22, ISSN: 1083-589X

Journal article

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

Consider a family of random ordinary differential equations on a manifolddriven by vector fields of the formk Ykαk (zt (ω)) where Yk are vector fields, isa positive number, zt is a 1L0 diffusion process taking values in possibly a differentmanifold, αk are annihilators of ker(L∗0). Under Hörmander type conditions on L0 weprove that, as approaches zero, the stochastic processes ytconverge weakly and inthe Wasserstein topologies. We describe this limit and give an upper bound for the rateof the convergence.

Journal article

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

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][0,1] and the hypoelliptic bridge is a continuous semi-martingale.

Journal article

Li X-M, 2016, Random perturbation to the geodesic equation, Annals of Probability, Vol: 44, Pages: 544-566, ISSN: 0091-1798

We study random “perturbation” to the geodesic equation. The geodesic equation is identified with a canonical differential equation on the orthonormal frame bundle driven by a horizontal vector field of norm 11. We prove that the projections of the solutions to the perturbed equations, converge, after suitable rescaling, to a Brownian motion scaled by 8n(n−1)8n(n−1) where nn is the dimension of the state space. Their horizontal lifts to the orthonormal frame bundle converge also, to a scaled horizontal Brownian motion.

Journal article

Chen X, Li X-M, 2014, Strong completeness for a class of stochastic differential equations with irregular coefficients, Electronic Journal of Probability, Vol: 19, ISSN: 1083-6489

We prove the strong completeness for a class of non-degenerate SDEs, whose coefficients are not necessarily uniformly elliptic nor locally Lipschitz continuous nor bounded.Moreover, for each p>0 there is a positive number T(p) such that for all t<T(p),the solution flow Ft(⋅) belongs to the Sobolev space W1,ploc. The main tool for this is the approximation of the associated derivative flow equations. As an application a differential formula is also obtained.

Journal article

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