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Longjie Xie

Publications and source records attributed to Longjie Xie.

At least 19 recordsLinked to original sources
arXiv2026

Infinite-horizon time-inhomogeneous Kolmogorov equation and non-autonomous multi-scale stochastic systems

We propose a unified approach, based on the infinite-horizon time-inhomogeneous Kolmogorov equation with a parameter, to study the asymptotic behavior of non-autonomous multi-scale stochastic systems with irregular coefficients, where both the fast and the slow equations depend on the highly oscillating time component. We establish both the weak and strong convergence of double averaging principle as well as the functional central limit theorem with new homogenized coefficients given in terms of the evolution system of invariant measures of the time-inhomogeneous fast frozen equation. Moreover, rates of convergence are obtained as byproducts of our argument, which do not depend on the regularities of the coefficients with respect to the time component and the fast variable.

math.PR↗
arXiv2026

Derivative estimates for SDEs with singular and unbounded coefficients

We develop a unified PDE-probabilistic framework for pointwise gradient and Hessian estimates of Markov semigroups associated with stochastic differential equations with singular and unbounded coefficients. Under mild local structural assumptions on the diffusion matrix and integrability/regularity conditions on the drift, we obtain quantitative sharp short-time regularization estimates as well as long-time decay bounds (including exponential and polynomial rates) for the first and second spatial derivatives of the semigroup. A distinctive feature of our results is the explicit dependence of these estimates on local norms of the coefficients (through scale-invariant quantities), without requiring any global smoothness, boundedness or uniform ellipticity. In particular, our approach allows for degenerate or highly irregular behavior at infinity, subject to suitable local ellipticity and Lyapunov/ergodicity controls. As applications, we establish solvability and regularity results for Poisson equations on the whole space with singular coefficients, and we derive pointwise gradient estimates for SDEs with distributional drifts via a Zvonkin-type transform.

math.PR↗
arXiv2026

Uniform-in-time diffusion approximations for multiscale stochastic systems

This paper establishes a quantitative, uniform-in-time diffusion approximation for the joint law of a broad class of fully coupled multiscale stochastic systems. We derive a precise characterization of the limiting joint distribution as a specific skew-product of the conditional equilibrium of the fast process and the homogenized law of the slow component, thereby providing a rigorous uniform-in-time formulation of the adiabatic elimination principle. The convergence rate explicitly separates the initial relaxation of the fast dynamics from the long-time homogenized evolution and depends only on the regularity of the coefficients in the slow variable. As a consequence, we obtain the first quantitative identification of the limiting stationary distribution of the original multiscale system and prove the commutativity of the limits $\eps\to0$ and $t\to\infty$ for a large class of observables. Our framework accommodates unbounded and irregular coefficients, degenerate structures, and weakly mixing dynamics. We illustrate its scope with three applications: {\it (i)} a uniform-in-time averaging principle for fast-slow systems; {\it (ii)} a uniform Smoluchowski--Kramers approximation for degenerate Langevin systems, yielding convergence of the joint position-scaled velocity law and global-in-time asymptotics of key thermodynamic functionals (e.g., total energy, entropy production, free energy); and {\it (iii)} the first uniform-in-time periodic homogenization result for SDEs with distributional drifts.

math.PR↗
arXiv2024

Asymptotic limit of fully coupled multi-scale non-linear stochastic system: the non-autonomous approximation method

In this paper, we develop a novel argument, the non-autonomous approximation method, to seek the asymptotic limits of the fully coupled multi-scale McKean-Vlasov stochastic systems with irregular coefficients, which, as summarized in [3,Section 7], remains an open problem in the field. We provide an explicit characterization for the averaged limit of the non-linear stochastic system, where both the choice of the frozen equation and the definition of the averaged coefficients are more or less unexpected since new integral terms with respect to the measure variable appear. More importantly, in contrast with the classical theory of multi-scale systems which focuses on the averaged limit of the slow process, we propose a new perspective that the asymptotic behavior of the entire system is actually governed by the limit of the fast motion. By studying the long-time estimates of the solution of the Kolmogorov equation in Wasserstein space, we identify the limiting distribution of the fast motion of the non-linear system, which, to the best of our knowledge, is new even for the classical multi-scale Itô SDEs. Furthermore, rates of convergence are also obtained, which are rather sharp and depend only on the regularity of the coefficients with respect to the slow variable. The innovation of our argument is to transform the non-linear system into a sequence of linear but non-autonomous systems, which is rather simple insofar as it avoids to involve the mean-field type PDEs associated with non-linear stochastic system, and at the same time, it turns out to be quite effective as it enables us to show that the strong convergence in the averaging principle of the non-linear stochastic system follows directly from the weak convergence, which significantly simplified the proof.

math.PR↗
arXiv2022

Functional law of large numbers and central limit theorem for slow-fast McKean-Vlasov equations

In this paper, we study the asymptotic behavior of a fully-coupled slow-fast McKean-Vlasov stochastic system. Using the non-linear Poisson equation on Wasserstein space, we first establish the strong convergence in the averaging principle of the functional law of large numbers type. In particular, the diffusion coefficient of the slow process can depend on the distribution of the fast motion. Then we consider the stochastic fluctuations of the original system around its average, and prove that the normalized difference will converge weakly to a linear McKean-Vlasov Ornstein-Uhlenbeck type process, which can be viewed as a functional central limit theorem. Extra drift and diffusion coefficients involving the expectation are characterized explicitly. Furthermore, the optimal rates of the convergence are also obtained.

math.PR↗
arXiv2022

Poisson equation on Wasserstein space and diffusion approximations for McKean-Vlasov equation

We consider the fully-coupled McKean-Vlasov equation with multi-time-scale potentials, and all the coefficients depend on the distributions of both the slow component and the fast motion. By studying the smoothness of the solution of the non-linear Poisson equation on Wasserstein space, we derive the asymptotic limit as well as the quantitative error estimate of the convergence for the slow process. Extra homogenized drift term containing derivative in the measure argument of the solution of the Poisson equation appears in the limit, which seems to be new and is unique for systems involving the fast distribution.

math.PR↗
arXiv2021

Asymptotic behavior of multiscale stochastic partial differential equations

In this paper, we study the asymptotic behavior of a semi-linear slow-fast stochastic partial differential equation with singular coefficients. Using the Poisson equation in Hilbert space, we first establish the strong convergence in the averaging principe, which can be viewed as a functional law of large numbers. Then we study the stochastic fluctuations between the original system and its averaged equation. We show that the normalized difference converges weakly to an Ornstein-Uhlenbeck type process, which can be viewed as a functional central limit theorem. Furthermore, rates of convergence both for the strong convergence and the normal deviation are obtained, and these convergence are shown not to depend on the regularity of the coefficients in the equation for the fast variable, which coincides with the intuition, since in the limit systems the fast component has been totally averaged or homogenized out.

math.PR↗
arXiv2021

Diffusion approximation for multi-scale stochastic reaction-diffusion equations

In this paper, we study the diffusion approximation for singularly perturbed stochastic reaction-diffusion equation with a fast oscillating term. The asymptotic limit for the original system is obtained, where an extra Gaussian term appears. Such a term is explicitly given in terms of the solution of Poisson equation in Hilbert space. Moreover, we also obtain the optimal rate of convergence, and the convergence rate is shown not to depend on the regularity of the coefficients of the original system with respect to the fast variable, which coincides with the intuition since the fast component has been totally homogenized out in the limit equation.

math.PR↗
arXiv2021

Averaging principle and normal deviations for multiscale stochastic systems

We study the asymptotic behavior for an inhomogeneous multiscale stochastic dynamical system with non-smooth coefficients. Depending on the averaging regime and the homogenization regime, two strong convergences in the averaging principle of functional law of large numbers type are established. Then we consider the small fluctuations of the system around its average. Nine cases of functional central limit type theorems are obtained. In particular, even though the averaged equation for the original system is the same, the corresponding homogenization limit for the normal deviation can be quite different due to the difference in the interactions between the fast scales and the deviation scales. We provide quite intuitive explanations for each case. Furthermore, sharp rates both for the strong convergences and the functional central limit theorems are obtained, and these convergences are shown to rely only on the regularity of the coefficients of the system with respect to the slow variable, and do not depend on their regularity with respect to the fast variable, which coincide with the intuition since in the limit equations the fast component has been totally averaged or homogenized out.

math.PR↗
arXiv2020

Averaging principle and normal deviations for multi-scale stochastic hyperbolic-parabolic equations

We study the asymptotic behavior of stochastic hyperbolic parabolic equations with slow and fast time scales. Both the strong and weak convergence in the averaging principe are established, which can be viewed as a functional law of large numbers. Then we study the stochastic fluctuations of the original system around its averaged equation. We show that the normalized difference converges weakly to the solution of a linear stochastic wave equation, which is a form of functional central limit theorem. We provide a unified proof for the above convergence by using the Poisson equation in Hilbert spaces. Moreover, sharp rates of convergence are obtained, which are shown not to depend on the regularity of the coefficients in the equation for the fast variable.

math.PR↗
arXiv2020

Diffusion approximation for fully coupled stochastic differential equations

We consider a Poisson equation in $\mathbb R^d$ for the elliptic operator corresponding to an ergodic diffusion process. Optimal regularity and smoothness with respect to the parameter are obtained under mild conditions on the coefficients. The result is then applied to establish a general diffusion approximation for fully coupled multi-time-scales stochastic differential equations with only Hölder continuous coefficients. Four different averaged equations as well as rates of convergence are obtained. Moreover, the convergence is shown to rely only on the regularities of the coefficients with respect to the slow variable, and does not depend on their regularities with respect to the fast component.

math.PR↗
arXiv2019

Superposition principle for non-local Fokker-Planck operators

We prove the superposition principle for probability measure-valued solutions to non-local Fokker-Planck equations, which in turn yields the equivalence between martingale problems for SDEs with jumps and such non-local PDEs with rough coefficients. As an application, we obtain a probabilistic representation for weak solutions of fractional porous media equations.

math.PR↗
arXiv2019

$L^q(L^p)$-theory of stochastic differential equations

In this paper we show the weak differentiability of the unique strong solution with respect to the starting point $x$ as well as Bismut-Elworthy-Li's derivative formula for the following stochastic differential equation in $\mathbb R^d$: $$ {\rm d} X_t=b(t,X_t){\rm d} t+σ(t,X_t){\rm d} W_t,\ \ X_0=x\in\mathbb R^d, $$ where $σ$ is bounded, uniformly continuous and nondegenerate, $\nablaσ\in \widetilde{\mathbb L}^{p_1}_{q_1}$ and $b\in \widetilde{\mathbb L}^{p_2}_{q_2}$ for some $p_i,q_i\in[2,\infty)$ with $\frac{d}{p_i}+\frac{2}{q_i}<1$, $i=1,2$, where $\widetilde{\mathbb L}^{p_i}_{q_i}, i=1,2$ are some localized spaces. Moreover, in the endpoint case $b\in \widetilde{\mathbb L}^{d; {\rm uni}}_\infty$, we also show the weak well-posedness.

math.PR↗
arXiv2019

Strong and weak convergence in the averaging principle for SDEs with Hölder coefficients

Using Zvonkin's transform and the Poisson equation in $R^d$ with a parameter, we prove the averaging principle for stochastic differential equations with time-dependent Hölder continuous coefficients. Sharp convergence rates with order $(α\wedge1)/2$ in the strong sense and $(α/2)\wedge1$ in the weak sense are obtained, considerably extending the existing results in the literature. Moreover, we prove that the convergence of the multi-scale system to the effective equation depends only on the regularity of the coefficients of the equation for the slow variable, and does not depend on the regularity of the coefficients of the equation for the fast component.

math.PR↗
arXiv2019

Blowup solutions of Grushin's operator

In this note, we consider the blowup phenomenon of Grushin's operator. By using the knowledge of probability, we first get expression of heat kernel of Grushin's operator. Then by using the properties of heat kernel and suitable auxiliary function, we get that the solutions will blow up in finite time.

math.AP↗