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Xue-Mei Li

Publications and source records attributed to Xue-Mei Li.

At least 55 records · Page 3Linked to original sources
arXiv2019

Concerning the geometry of stochastic differential equations and stochastic flows

Following Le Jan and Watanabe we define a connection associated with a non-degenrrate diffusion operators. This connection is characterized here and shown to be the Levi-Civita connection for gradient systems. This both explains why such systems have useful properties and allows us to extend these properties to more general systems. Topics described here include: moment estimates for $Tξ_t$, a Weitzenböck formula for the generator of the semigroup on p-forms induced by the flow, a Bismut type formula for $d\log p_t$ in terms of an arbitrary metric connection, and a generalized Bochner vanishing theorem. A comprehensive theory on this, its generalization to semi-elliptic case and applications is published in the book `On the geometry of diffusion operators and stochastic flows'. Related to this is also the book `The geometry of filtering'. This article is easier to read.

math.PR↗
arXiv2019

On Extensions of Myers' Theorem

Let $M$ be a compact Riemannian manifold and $h$ a smooth function on $M$. Let $ρ^h(x)=\inf_{|v|=1}\left(Ric_x(v,v)-2Hess(h)_x(v,v) \right)$. Here $Ric_x$ denotes the Ricci curvature at $x$ and $Hess(h)$ is the Hessian of $h$. Then $M$ has finite fundamental group if $Δ^h-ρ^h<0$. Here $Δ^h=: Δ+2L_{\nabla h}$ is the Bismut-Witten Laplacian. This leads to a quick proof of recent results on extension of Myers' theorem to manifolds with mostly positive curvature. There is also a similar result for noncompact manifolds.

math.DG↗
arXiv2019

Strong p-completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds

We introduce strong p-completeness and use them for studying the continuous dependence of solutions of SDE's on non-compact manifolds. We obtain conditions for the existence of global smooth solution flow, and prove their diffeomorphism properties. The criterion is in terms of the growth of the solutions of the linearized equations (i.e. the derivative flows). It is also given in terms of the driving vector fields and their derivatives. For the Euclidean spaces there are three principle classes of examples: (1) the driving vector fields grow at most $|x|^{1-2α}$ and the derivatives of the driving vector are allowed to be the order $|x|^α$; (2) the coefficients of the SDE grows linearly while its derivative allowed to grow logarithmically, (3) The coefficients are bounded while the derivatives are allowed to grow linearly. We also given test on non-explosions.

math.PR↗
arXiv2019

Averaging dynamics driven by fractional Brownian motion

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

math.PR↗
arXiv2019

Perturbation of Conservation Laws and Averaging on Manifolds

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 ${\mathcal L}_x$ 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 ${\mathcal L}_x$ satisfies Hörmander's bracket conditions, or more generally ${\mathcal L}_x$ 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.

math.PR↗
arXiv2018

Generalised Brownian bridges: examples

We observe that the probability distribution of the Brownian motion with drift $-c \frac x {1-t}$ where $c\not =1$ is singular with respect to that of the classical Brownian bridge measure on $[0,1]$, while their Cameron-Martin spaces are equal set-wise if and only if $c> \frac 12$, 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.

math.PR↗
arXiv2018

Homogenisation On Homogeneous Spaces

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 $ε>0$, interpolating hypo-elliptic diffusions on $H$ and translates of exponential maps on $G$ and examine the dynamics as $ε\to 0$. 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 $\frac 1 ε$), 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 ${\mathcal L}^ε=\frac 1 ε \sum_k (A_k)^2+ \frac 1ε A_0+ Y_0$ where $Y_0, A_k$ are left invariant vector fields and $\{A_k\}$ generate the Lie-algebra of $H$.

math.PR↗
arXiv2017

First Order Feynman-Kac Formula

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

math.PR↗
arXiv2017

Doubly Damped Stochastic Parallel Translations and Hessian Formulas

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 $C(\frac 1 t +\frac {d^2}{t^2})$ 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.

math.PR↗
arXiv2017

Strict local martingales: examples

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

math.PR↗
arXiv2017

Reflected brownian motion: selection, approximation and linearization

We construct a family of SDEs whose solutions select a reflected Brownian flow as well as a stochastic damped transport process (W\_t). The latter gives a representation for the solutions to the heat equation for differential 1-forms with the absolute boundary conditions; it evolves pathwise by the Ricci curvature in the interior, by the shape operator on the boundary and driven by the boundary local time, and has its normal part erased on the boundary. 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 this agrees with the construction of N. Ikeda and S. Watanabe \cite{Ikeda-Watanabe} by Poisson point processes. This leads also to an approximation for the boundary local time in the topology of uniform convergence; not in the semi-martingale topology, indicating the difficulty for the convergence of solutions of a family of random ODE's, with nice coefficients, to the solution of an equation with jumps and driven by the local time. In addition, We note that (W\_t) is the weak derivative of a family of reflected Brownian motions with respect to the starting point.

math.PR↗
arXiv2016

Hessian formulas and estimates for parabolic Schrödinger operators

We study the Hessian of the fundamental solution to the parabolic problem for weighted Schrödinger operators of the form $\frac 12 Δ+\nabla h-V$ proving a second order Feynman-Kac formula and obtaining Hessian estimates. For manifolds with a pole, we use the Jacobian determinant of the exponential map to offset the volume growth of the Riemannian measure and use the semi-classical bridge as a delta measure at $y_0$ to obtain exact Gaussian estimates. These estimates are in terms of bounds on $Ric-2 Hess (h)$, on the curvature operator, and on the cyclic sum of the gradient of the Ricci tensor.

math.PR↗
arXiv2016

Limits of Random Differential Equations on Manifolds

Consider a family of random ordinary differential equations on a manifold driven by vector fields of the form $\sum_kY_kα_k(z_t^ε(ω))$ where $Y_k$ are vector fields, $ε$ is a positive number, $z_t^ε$ is a ${1\over ε} {\mathcal L}_0$ diffusion process taking values in possibly a different manifold, $α_k$ are annihilators of $ker ({\mathcal L}_0^*)$. Under Hörmander type conditions on ${\mathcal L}_0$ we prove that, as $ε$ approaches zero, the stochastic processes $y_{t\over ε}^ε$ converge weakly and in the Wasserstein topologies. We describe this limit and give an upper bound for the rate of the convergence.

math.PR↗
arXiv2016

Random perturbation to the geodesic equation

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 $1$. We prove that the projections of the solutions to the perturbed equations, converge, after suitable rescaling, to a Brownian motion scaled by ${\frac{8}{n(n-1)}}$ where $n$ is the dimension of the state space. Their horizontal lifts to the orthonormal frame bundle converge also, to a scaled horizontal Brownian motion.

math.PR↗
arXiv2015

On Hypoelliptic Bridge

We prove that if the Markov generator of a diffusion process satisfies the two step strong Hörmander condition, the conditioned hypoelliptic bridge satisfies an integral bound and is a continuous semi-martingale.

math.PR↗
arXiv2014

Strong completeness for a class of stochastic differential equations with irregular coefficients

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 $t$, the solution flow $F_t$ is weakly differentiable and for each $p>0$ there is a positive number $T(p)$ such that for all $t<T(p)$, the solution flow $F_t(\cdot)$ belongs to the Sobolev space $W_{\loc}^{1,p}$. The main tool for this is the approximation of the associated derivative flow equations. As an application a differential formula is also obtained.

math.PR↗
arXiv2012

Effective Diffusions with Intertwined Structures

Let p:N->M be a surjective map of smooth manifolds. We are concerned with singular perturbation problems associated to a pair of second order positive definite differential operators with no zero order terms, that are intertwined by p. We discuss the associated random perturbations of stochastic differential equations and present a number of examples including perturbation to geodesic flows and construction of a Brownian motion on S^2 through homogenisation of SDE's on the Hopf fibration.

math.PR↗