arXiv subjects
Publications and source records attributed to Xue-Mei Li.
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.
We investigate the effective behaviour of a small transversal perturbation of order $ε$ to a completely integrable stochastic Hamiltonian system, by which we mean a stochastic differential equation whose diffusion vector fields are formed from a completely integrable family of Hamiltonian functions $H_i, i=1,\dots n$. An averaging principle is shown to hold and the action component of the solution converges, as $ε\to 0$, to the solution of a deterministic system of differential equations when the time is rescaled at $1/ε$. An estimate for the rate of the convergence is given. In the case when the perturbation is a Hamiltonian vector field, the limiting deterministic system is constant in which case we show that the action component of the solution scaled at $1/ε^2$ converges to that of a limiting stochastic differentiable equation.
I was asked to make my, by now quite old PhD thesis, available on the arxiv, for parts of it was never submitted for publication. The thesis offers a systematic study of stochastic differential equations (SDEs) on non-compact spaces. In particular we solve the open problem on strong completeness. An SDE is strongly complete if its solution can be chosen to depend continuously in space and in time for all time. The question is whether non-explosion, with possibly additional assumptions, implies strong completeness. Strong completeness of an SDE implies that its solution depends continuously on the initial condition, opening up possibility for numerical solutions, and the existence of a perfect Cocycle (a basic assumption on random dynamical systems). This was known only for compact manifolds and for linear state spaces, methods for either are not applicable to a general space. We also obtain existence of the global smooth solution flow of SDEs on $R^n$ (sometimes allowing substantial growth of the derivative of the coefficients, removing the global Lipschitz continuity condition). Non-explosion, the $C_0$-property, and the derivative flow are studied. We showed Bismut-Witten Laplacians are essentially self-adjojnt, paving the way for studying theirs semigroups acting on functions and on differential forms. We relate the Markovian semi-group on differential one forms with the semi-group $P_t$ on functions (inter-twining), find a method for proving path integration formulas for $dP_tf$, path integration formula for semi-group on differential forms, moment bounds for the derivative flows. Relation are obtained between intrinsic topological and geometrical properties of the manifold and that of SDEs. Information on the homotopy and cohomology of the manifolds are obtained from moment stability of the stochastic flows.
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 noise and providing new examples. The emphasise of the review will be on the recently developed effective dynamic theory for two scale random systems with fractional noise: Stochastic Averaging and `Rough Diffusion Homogenisation Theory'. We also study the geometric models with perturbations to symmetries.
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 $L^2$ 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.
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.
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.
Given a pair of second order diffusion operators, one on the total space of a principle bundle $N$ and the other on the base space $M$, intertwined by the projection $π:N\to M$, if the operator ${\mathcal A}$ on the base manifold has constant rank, we define a semi-connection on the principal bundle which allows to split the diffusion operator ${\mathcal B}$ on the total space into the sum of the horizontal lift of ${\mathcal A}$ and the other vertical. This allow to conclude a disintegration theorem for the law of ${\mathcal B}$. As an application, a decomposition of stochastic flow is given.
We prove a Kodaira-Hodge decomposition on differential 1-forms on the space of non-smooth paths over a Riemannian manifold, allowing us to define the corresponding first cohomology group. This uses the Itô map of a Brownian system and damped stochastic parallel translation.
An integration by parts formula is the foundation for stochastic analysis on path spaces over a (finite dimensional) Riemannian manifold or over $R^n$, from which we may deduce the operator $d$ is closable and define the Laplacian operator on path spaces. A useful formula on the Riemannian manifold is $$dP_tf(v)=(1/t)E f(x_t) \int_0^t \langle d\{x_s\}, v_s\rangle ,$$ for $P_t$ the heat semi-group, $x_t$ the BM, $v_t$ the derivative flow or its conditional expectation (which is a damped parallel translation), $d\{x_s\}$ is the martingale part of $x_t$. As a meta theorem, this leads to the Clark-Ocone formula (martingale representation theorem with specific integrand) and Logrithmic Sobolev inequalities. Interpreted appropriately, the latter formula is obviously a special case of the integration by parts formula. Here we show by the Markov property and by induction that the latter formula implies the integration by parts formula. WE also use Bismut's original approach to prove an integration by parts formula, using a connection with torsion and one on the free path space.
Integration by parts formulae are given for a class of measures on the space of paths of a smooth manifold $M$ determined by the laws of degenerate diffusions. The mother of such formulae, on the path space of diffeomorphism group of $M$ is shown to arise from a quasi-invariance property of measures determined by stochastic flows. From this the other formulae are derived by filtering out redundant noise using an associated LeJan-Watanabe connection.
There are two open problem on the analysis of continuous paths on a Riemannian manifold, the Markov uniqueness and the independence of the closure of the differential operator $d$ on its initial domain. The operator $d$ acts naturally on $BC^1$ functions, one is concerned with its extensions to the $L^2$ spaces. With a suitable choice of an initial domain we denote by $D^{2,1}$ its closure under the graph norm. For the Wiener space, the domain of $d$ can be classified, as a consequence its extension is unique whether the initial domain is smooth cylindrical or in $BC^1$ etc. This has not shown to be the same when the measure is the probability distribution of any smooth elliptic diffusion. In an earlier paper, we have shown that the closure of $BC^\infty$ functions agree with that of smooth cylindrical functions, leaving an undesirable gap. The Markov uniqueness is essentially concerned with the problem whether there exists a unique Markov process on the path space whose Markov generator agrees with the infinite-dimensional Laplacian on $C^\infty$ cylindrical functions. Here we reduce Markov uniqueness to whether the pull back of $D^{2,1}$ by the ito map is $ D^{2,1}$ (i.e. a surjection). We also propose a possible approach for tackle this problem.
We use a basic martingale method to show a differentiation formula for the derivatives $$d(P_tf)(x_0)(v_0)={1\over t} E f(x_t) \int_0^t \langle Y(x_s)(v_s),dB_t\rangle_{R^m}.$$ These are proved first on $R^n$, then on manifolds. Afterwards for solutions of heat equations on differential forms, and a second order formula.
Formulae are given for $dP_t ϕ$, $d^*P_tϕ$ and $ΔP_tϕ$ for $P_t$ the heat semigroup acting on a q-form $ϕ$. The formulae are Brownian motion expectations of $ϕ$ composed with random translations determined by Weitzenbock curvarure terms. Derivatives of the curvature are not involved.
An approach to analysis on path spaces of Riemannian manifolds is described. The spaces are furnished with `Brownian motion' measure which lies on continuous paths, though differentiation is restricted to directions given by tangent paths of finite energy. An introduction describes the background for paths on ${\mathbb R}^m$ and Malliavin calculus. For manifold valued paths the approach is to use `Itô' maps of suitable stochastic differential equations as charts . `Suitability' involves the connection determined by the stochastic differential equation. Some fundamental open problems concerning the calculus and the resulting `Laplacian' are described. A theory for more general diffusion measures is also briefly indicated. The same method is applied as an approach to getting over the fundamental difficulty of defining exterior differentiation as a closed operator, with success for one \& two forms leading to a Hodge -Kodaira operator and decomposition for such forms. Finally there is a brief description of some related results for loop spaces.
A great open problem is: can one learn the topology of the non-smooth path spaces with an L2 Hodge-deRham theory This one hopes to establish through a suitable complex of differential forms. Since the space is a Banach manifolds, and the Hodge theory is based on Hilbert spaces, it is trick to find such a complex. The relatively simpler Bismut tangent spaces and their tensor products, whose dual spaces are natural candidates for the complex, cannot be used, because these spaces are not necessarily closed under the Lie bracket operation if there is the effect of curvature. In this article we seek out a class of nice vector fields whose brackets behaves are calculable and behave nicely the damped tensor vector fields, using an Itô mp, which is also used by the authors in `Special Itô maps and an L2 Hodge theory for one forms on path spaces. Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999), 145-162, CMS Conf. Proc., 28, Amer. Math. Soc., Providence, RI, 2000').
A unified treatment is given of some results of H. Donnelly-P. Li and L. Schwartz concerning the behaviour of heat semigroups on open manifolds with given compactifications, on one hand, and the relationship with the behaviour at infinity of solutions of related stochastic differential equations on the other. A principal tool is the use of certain covers of the manifold: which also gives a non-explosion test. As a corollary we obtain known results about the behaviour of Brownian motions on a complete Riemannian manifold with Ricci curvature decaying at most quadratically in the distance function.
In this paper we discuss the stability of stochastic differential equations and the interplay between the moment stability of a SDE and the topology of the underlying manifold. Sufficient and necessary conditions are given for the moment stability of a SDE in terms of the coefficients. Finally we prove a vanishing result for the fundamental group of a complete Riemannian manifold in terms of purely geometrical quantities.