## Publications

60 results found

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

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.

Hairer M, Li X-M, Averaging dynamics driven by fractional Brownian motion, *Annals of Probability*, ISSN: 0091-1798

We consider slow / fast systems where the slow system is driven by fractionalBrownian motion with Hurst parameterH >12. We show that unlike in the caseH=12, convergence to the averaged solution takes place in probability and thelimiting process solves the ‘naïvely’ averaged equation. Our proof strongly relieson the recently obtained stochastic sewing lemma.

Gozlan N, Li X-M, Madiman M, et al., 2019, Log-Hessian formula and the Talagrand conjecture, Publisher: arXiv

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.

Li X-M, 2018, Perturbation of Conservation Laws and Averaging on Manifolds, *Computation and Combinatories in Dynamics, stochastic and control, The Abel Symposium, Rosendal, Norway, August 2016*

We prove a stochastic averaging theorem for stochastic differential equationsin which the slow and the fast variables interact. The approximate Markov fastmotion is a family of Markov process with generator ${\mathcal L}_x$. Thetheorem is proved under the assumption that ${\mathcal L}$ satisfiesH\"ormander's bracket conditions, or more generally ${\mathcal L}$ is a familyof Fredholm operators with sub-elliptic estimates. On the other hand aconservation law of a dynamical system can be used as a tool for separating thescales in singular perturbation problems. We discuss a number of motivatingexamples from mathematical physics and from geometry where we use non-linearconservation laws to deduce slow-fast systems of stochastic differentialequations.

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.

Hairer X, Generalised Brownian bridges: examples, *Markov Processes and Related Fields*, ISSN: 1024-2953

We observe that the probability distribution of the Brownian motionwith drift−cx/(1−t) wherec6= 1 is singular with respect to that of the classicalBrownian bridge measure on [0,1], while their Cameron – Martin spaces areequal set-wise if and only ifc >1/2, providing also examples of exponentialmartingales on [0,1) not extendable to a continuous martingale on [0,1]. Otherexamples of generalised Brownian bridges are also studied.

Li X-M, 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.

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 $\epsilon>0$, 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 $H$.

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.

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.

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

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.

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.

Li X-M, 2015, Limits of random differential equations on manifolds, *Probability Theory and Related Fields*, Vol: 166, Pages: 659-712, ISSN: 1432-2064

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.

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.

Li X-M, 2012, The stochastic differential equation approach to analysis on path space, New trends in stochastic analysis and related topics, Publisher: World Sci. Publ., Hackensack, NJ, Pages: 207-226

Li X-M, Scheutzow M, 2011, Lack of strong completeness for stochastic flows, *Annals of Probability*, Vol: 39, Pages: 1407-1421, ISSN: 0091-1798

It is well known that a stochastic differential equation (SDE) on a Euclidean space driven by a Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. When the coefficients are only locally Lipschitz, then a maximal continuous flow still exists but explosion in finite time may occur. If, in addition, the coefficients grow at most linearly, then this flow has the property that for each fixed initial condition x, the solution exists for all times almost surely. If the exceptional set of measure zero can be chosen independently of x, then the maximal flow is called strongly complete. The question, whether an SDE with locally Lipschitz continuous coefficients satisfying a linear growth condition is strongly complete was open for many years. In this paper, we construct a two-dimensional SDE with coefficients which are even bounded (and smooth) and which is not strongly complete thus answering the question in the negative.

Li X-M, 2011, Intertwinned diffusions by examples, Stochastic analysis 2010, Publisher: Springer, Heidelberg, Pages: 51-71

Chen X, Li X-M, Wu B, 2010, A concrete estimate for the weak Poincaré inequality on loop space, *Probability Theory and Related Fields*, Vol: 151, Pages: 559-590, ISSN: 1432-2064

The aim of the paper is to study the pinned Wiener measure on the loop space over a simply connected compact Riemannian manifold together with a Hilbert space structure and the Ornstein–Uhlenbeck operator d*d. We give a concrete estimate for the weak Poincaré inequality, assuming positivity of the Ricci curvature of the underlying manifold. The order of the rate function is s−α for any α > 0.

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

We show that the Laplacian on the loop space over a class of Riemannian manifolds has a spectral gap. The Laplacian is defined using the Levi-Civita connection, the Brownian bridge measure and the standard Bismut tangent spaces.

Elworthy KD, Jan YL, Li X-M, 2010, Diffusion Operators, The Geometry of Filtering, Publisher: Springer Basel, Pages: 1-10, ISBN: 9783034601757

Elworthy KD, Jan YL, Li X-M, 2010, Example: Stochastic Flows, The Geometry of Filtering, Publisher: Springer Basel, Pages: 121-133, ISBN: 9783034601757

Elworthy KD, Le Jan Y, Li X-M, 2010, The geometry of filtering, Publisher: Birkhäuser Verlag, Basel, ISBN: 978-3-0346-0175-7

Chen X, Li X-M, Wu B, 2010, A spectral gap for the Brownian bridge measure on hyperbolic spaces, Progress in analysis and its applications, Publisher: World Sci. Publ., Hackensack, NJ, Pages: 398-404

Elworthy KD, Jan YL, Li X-M, 2010, Decomposition of Diffusion Operators, The Geometry of Filtering, Publisher: Springer Basel, Pages: 11-32, ISBN: 9783034601757

Elworthy KD, Jan YL, Li X-M, 2010, Appendices, The Geometry of Filtering, Publisher: Springer Basel, Pages: 135-158, ISBN: 9783034601757

Elworthy KD, Jan YL, Li X-M, 2010, The Commutation Property, The Geometry of Filtering, Publisher: Springer Basel, Pages: 101-114, ISBN: 9783034601757

Li X-M, 2008, An averaging principle for a completely integrable stochastic Hamiltonian system, *Nonlinearity*, Vol: 21, Pages: 803-822, ISSN: 0951-7715

Elworthy KD, Li X-M, 2008, An $L^2$ theory for differential forms on path spaces. I, *Journal of Functional Analysis*, Vol: 254, Pages: 196-245, ISSN: 0022-1236

Elworthy KD, Li X-M, 2007, Itô maps and analysis on path spaces, *Mathematische Zeitschrift*, Vol: 257, Pages: 643-706, ISSN: 0025-5874

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