Multi-scale homogenization with bounded ratios and Anomalous Slow Diffusion

Rate of homogenization for fully-coupled McKean-Vlasov SDEs

We consider a fully-coupled slow-fast system of McKean-Vlasov SDEs with full dependence on the slow and fast component and on the law of the slow component and derive convergence rates to its homogenized limit. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. In the course of the proof we obtain related ergodic theorems and we gain results on the regularity of Poisson type of equations and of the associated Cauchy-Problem on the Wasserstein space that are of independent interest.

具有Neumann边界条件的偏微分方程的定量均匀化：概率方法

Quantitative Homogenization of PDEs with Neumann boundary conditions: a probabilistic approach

本文通过研究反射慢快系统倒向随机微分方程的定量均匀化，对凸域上具有Neumann边界条件的多尺度半线性二阶偏微分方程的粘性解进行定量均匀化分析。

Limiting distribution of elliptic homogenization error with periodic diffusion and random potential

We study the limiting probability distribution of the homogenization error for second order elliptic equations in divergence form with highly oscillatory periodic conductivity coefficients and highly oscillatory stochastic potential. The effective conductivity coefficients are the same as those of the standard periodic homogenization, and the effective potential is given by the mean. We show that in the short range correlation setting, the limiting distribution of the random part of the homogenization error, as random elements in proper Hilbert spaces, is Gaussian and can be characterized by t…

Maximum Likelihood Drift Estimation for Multiscale Diffusions

We study the problem of parameter estimation using maximum likelihood for fast/slow systems of stochastic differential equations. Our aim is to shed light on the problem of model/data mismatch at small scales. We consider two classes of fast/slow problems for which a closed coarse-grained equation for the slow variables can be rigorously derived, which we refer to as averaging and homogenization problems. We ask whether, given data from the slow variable in the fast/slow system, we can correctly estimate parameters in the drift of the coarse-grained equation for the slow variable, using maximu…

Quantitative estimates in stochastic homogenization for correlated coefficient fields

This paper is about the homogenization of linear elliptic operators in divergence form with stationary random coefficients that have only slowly decaying correlations. It deduces optimal estimates of the homogenization error from optimal growth estimates of the (extended) corrector. In line with the heuristics, there are transitions at dimension $d=2$, and for a correlation-decay exponent $β=2$; we capture the correct power of logarithms coming from these two sources of criticality. The decay of correlations is sharply encoded in terms of a multiscale logarithmic Sobolev inequality (LSI) for t…

对由维纳或赫尔米特过程驱动的均质极限进行统计推断

Statistical Inference for Homogenization Limits Driven by Wiener or Hermite Processes

研究慢-快系统均质极限下的扩散系数和Hurst参数估计，通过维纳或赫尔米特过程驱动的随机微分方程，利用重正化二次变分和Wiener展开证明估计器的一致性及非中心极限定理。

有限状态马尔可夫博弈中多智能体学习动力学的均匀化

Homogenization of Multi-agent Learning Dynamics in Finite-state Markov Games

本文提出通过降低学习率和提高更新频率重标度学习过程，将智能体参数视为慢变量，证明了多智能体在有限状态马尔可夫博弈中的学习动力学收敛于常微分方程，提供了确定性的近似方法。

Moderate deviations principle for systems of slow-fast diffusions

In this paper, we prove the moderate deviations principle (MDP) for a general system of slow-fast dynamics. We provide a unified approach, based on weak convergence ideas and stochastic control arguments, that cover both the averaging and the homogenization regimes. We allow the coefficients to be in the whole space and not just the torus and allow the noises driving the slow and fast processes to be correlated arbitrarily. Similar to the large deviation case, the methodology that we follow allows construction of provably efficient Monte Carlo methods for rare events that fall into the moderat…

抛物方程中长程随机势的分数同调

Fractional Homogenization of Parabolic Equations with Long-Range Random Potentials

本文研究了一维抛物方程在长程相关随机势下的分数同调理论，揭示了由长程依赖驱动的新同调机制，结合随机同调、分数微积分和异常扩散理论。

Averaging for SDE-BSDE with null recurrent fast component Application to homogenization in a non periodic media

We establish an averaging principle for a family of solutions$(X^{\varepsilon}, Y^{\varepsilon})$ $ :=$ $(X^{1,\,\varepsilon},\,X^{2,\,\varepsilon},\, Y^{\varepsilon})$ of a system of SDE-BSDEwith a null recurrent fast component $X^{1,\,\varepsilon}$. Incontrast to the classical periodic case, we can not rely on aninvariant probability and the slow forward component$X^{2,\,\varepsilon}$ cannot be approximated by a diffusion process.On the other hand, we assume that the coefficients admit a limit in aCesaro sense. In such a case, the limit coefficients may havediscontinuity. We show that we can…

具有状态依赖切换的慢-快随机微分方程的扩散逼近

Diffusion Approximation for Slow-Fast SDEs with State-Dependent Switching

本文研究具有状态依赖切换的慢-快随机微分方程的扩散逼近，利用鞅问题方法和泊松方程识别弱极限过程，并证明弱收敛阶为1/2且该阶是最优的。

Discrete-Time Statistical Inference for Multiscale Diffusions

We study statistical inference for small-noise-perturbed multiscale dynamical systems under the assumption that we observe a single time series from the slow process only. We construct estimators for both averaging and homogenization regimes, based on an appropriate misspecified model motivated by a second-order stochastic Taylor expansion of the slow process with respect to a function of the time-scale separation parameter. In the case of a fixed number of observations, we establish consistency, asymptotic normality, and asymptotic statistical efficiency of a minimum contrast estimator (MCE),…