Publications
71 results found
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.
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, 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, 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
Li X-M, 2008, An averaging principle for a completely integrable stochastic Hamiltonian system, Nonlinearity, Vol: 21, Pages: 803-822, ISSN: 0951-7715
Li X-M, Elworthy KD, 2007, Geometric stochastic analysis on path spaces, the International Congress of Mathematicians
Elworthy KD, Li X-M, 2007, Itô maps and analysis on path spaces, Mathematische Zeitschrift, Vol: 257, Pages: 643-706, ISSN: 0025-5874
Elworthy KD, Li X-M, 2006, Geometric stochastic analysis on path spaces, International Congress of Mathematicians. Vol. III, Publisher: Eur. Math. Soc., Zürich, Pages: 575-594
Elworthy KD, Li X-M, 2006, Intertwining and the Markov uniqueness problem on path spaces, Stochastic partial differential equations and applications—VII, Publisher: Chapman & Hall/CRC, Boca Raton, FL, Pages: 89-95
Arnaudon M, Li X-M, 2005, Barycenters of measures transported by stochastic flows, The Annals of Probability, Vol: 33, Pages: 1509-1543, ISSN: 0091-1798
Arnaudon M, Hairer X, 2004, Barycentres of probability measures transported by stochastic flows, Annals of Probability, ISSN: 0091-1798
Elworthy KD, Le Jan Y, Li X-M, 2004, Equivariant diffusions on principal bundles, Stochastic analysis and related topics in Kyoto, Publisher: Math. Soc. Japan, Tokyo, Pages: 31-47
Li XUE-MEI, Elworthy KD, Le Jan Y, 2003, Equivariant Diffusions on Principal Bundles, Advanced Studies in Pure Mathematics, 41
Elworthy KD, Li X-M, 2003, Some families of $q$-vector fields on path spaces, Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol: 6, Pages: 1-27, ISSN: 0219-0257
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.
Li X-M, Wang F-Y, 2003, On the compactness of manifolds, Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol: 6, Pages: 29-38, ISSN: 0219-0257
Elworthy KD, Li X-M, 2003, Gross-Sobolev spaces on path manifolds: uniqueness and intertwining by Itô maps, Comptes Rendus Mathématique. Académie des Sciences. Paris, Vol: 337, Pages: 741-744, ISSN: 1631-073X
Elworthy KD, Li X-M, 2000, Special Itô maps and an $L^2$ Hodge theory for one forms on path spaces, Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999), Publisher: Amer. Math. Soc., Providence, RI, Pages: 145-162
Elworthy KD, Le Jan Y, Li X-M, 1999, On the geometry of diffusion operators and stochastic flows, Publisher: Springer-Verlag, Berlin, ISBN: 3-540-66708-3
Arnaudon M, Li X-M, Thalmaier A, 1999, Manifold-valued martingales, changes of probabilities, and smoothness of finely harmonic maps, Annales de l’Institut Henri Poincaré. Probabilités et Statistiques, Vol: 35, Pages: 765-791, ISSN: 0246-0203
Elworthy KD, Li X-M, Yor M, 1999, The importance of strictly local martingales; applications to radial Ornstein-Uhlenbeck processes, Probability Theory and Related Fields, Vol: 115, Pages: 325-355, ISSN: 0178-8051
Elworthy KD, Li X-M, 1998, Bismut type formulae for differential forms, Comptes Rendus de l’Académie des Sciences. Série I. Mathématique, Vol: 327, Pages: 87-92, ISSN: 0764-4442
Elworthy KD, Li X-M, Rosenberg S, 1998, Bounded and $L^2$ harmonic forms on universal covers, Geometric and Functional Analysis, Vol: 8, Pages: 283-303, ISSN: 1016-443X
Elworthy KD, Li XM, Yor M, 1997, On the tails of the supremum and the quadratic variation of strictly local martingales, Séminaire de Probabilités, XXXI, Publisher: Springer, Berlin, Pages: 113-125
Elworthy KD, Le Jan Y, Li X-M, 1997, Concerning the geometry of stochastic differential equations and stochastic flows, New Trends in Stochastic Analysis: Proceedings of a Tanaguchi International Workshop Charingworth Manor September 21-27 1994, Publisher: World Scientific, Pages: 107-131
Le Jan and Watanabe showed that a non-degenerate stochastic flow {ξt : t ≥0} on a manifold M determines a connection on M. This connection is characterized here and shown to be the Levi-Civita connection for gradient systems. Thisboth explains why such systems have useful properties and allows us to extendthese properties to more general systems. Topics described here include: momentestimates for T ξt, a Weitzenbock formula for the generator of the semigroup on ¨p-forms induced by the flow, a Bismut type formula for d log pt in terms of anarbitrary metric connection, and a generalized Bochner vanishing theorem
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