Periodic homogenization with an interface: the one-dimensional case

对称Lévy型算子的周期均匀化的高阶收敛率

High-order convergence rates of periodic homogenization for symmetric Lévy type operators

本文针对对称Lévy型算子，涵盖次临界α-稳定、临界和超临界扩散区域，建立周期均匀化的高阶收敛率，通过分解跳跃核在不同空间尺度的贡献实现。

Periodic Homogenization of a Lévy-Type Process with Small Jumps

In this article, we consider the problem of periodic homogenization of a Feller process generated by a pseudo-differential operator, the so-called Lévy-type process. Under the assumptions that the generator has rapidly periodically oscillating coefficients, and that it admits "small jumps" only (that is, the jump kernel has finite second moment), we prove that the appropriately centered and scaled process converges weakly to a Brownian motion with covariance matrix given in terms of the coefficients of the generator. The presented results generalize the classical and well-known results related…

带半透界面的多变量扩散的均匀化

Homogenization of a multivariate diffusion with semipermeable interfaces

针对带斜穿透半透超平面界面的多变量扩散均匀化问题，证明带局部时间项的随机微分方程系统存在唯一弱解，得到界面间距趋于零时的弱极限，揭示奇异项消失并产生界面诱导正则漂移的规律。

Large time average of reachable sets and Applications to Homogenization of interfaces moving with oscillatory spatio-temporal velocity

We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be reformulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the sublinear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.

Homogenization of periodic diffusion with small jumps

In this paper, we study the homogenization of a diffusion process with jumps, that is, Feller process generated by an integro-differential operator. This problem is closely related to the problem of homogenization of boundary value problems arising in studying the behavior of heterogeneous media. Under the assumptions that the corresponding generator has vanishing drift coefficient, rapidly periodically oscillating diffusion and jump coefficients, that it admits only "small jumps" (that is, the jump kernel has finite second moment) and under certain additional regularity conditions, we prove t…

Auxiliary SDEs for homogenization of quasilinear PDEs with periodic coefficients

We study the homogenization property of systems of quasi-linear PDEs of parabolic type with periodic coefficients, highly oscillating drift and highly oscillating nonlinear term. To this end, we propose a probabilistic approach based on the theory of forward-backward stochastic differential equations and introduce the new concept of ``auxiliary SDEs.''

Periodic homogenization with an interface

We consider a diffusion process with coefficients that are periodic outside of an 'interface region' of finite thickness. The question investigated in the articles [1,2] is the limiting long time / large scale behaviour of such a process under diffusive rescaling. It is clear that outside of the interface, the limiting process must behave like Brownian motion, with diffusion matrices given by the standard theory of homogenization. The interesting behaviour therefore occurs on the interface. Our main result is that the limiting process is a semimartingale whose bounded variation part is proport…

晶格 ${\mathbf Z}^d$ 上具有周期漂移系数的扩散的同质化；对数 Sobolev 不等式的应用

Homogenization of diffusions on the lattice ${\mathbf Z}^d$ with periodic drift coefficients; Application of logarithmic Sobolev inequality

利用基于对数 Sobolev 不等式的无穷维扩散过程的均匀遍历定理，证明了从状态空间中关于不变测度几乎处处任一点出发的过程的同质化性质，并解释为具有随机系数的无穷维扩散同质化问题的解。

切换扩散过程的周期均匀化

Periodic Homogenization for Switching Diffusions

本文研究一类切换扩散过程的周期均匀化技术，在一致椭圆性与不可约性假设下，刻画了过程在扩散尺度下的极限波动，并计算出由切换引起的极限扩散率的额外贡献。

Homogenization of a periodic semilinear elliptic degenerate PDE

In this paper a semilinear elliptic PDE with rapidly oscillating coefficients is homogenized. The novetly of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some portion of $\R^d$.\\ Our fully probabilistic method is based on the connection between PDEs and BSDEs with random terminal time and the weak convergence of a class of diffusion processes.

具有局部时间的粘性随机微分方程的极限定理及其在随机正则化中的应用

Limit theorems for sticky SDEs with local times and applications to stochastic homogenization

本文研究了具有局部时间的粘性随机微分方程的极限定理，揭示了在存在半渗透界面时扩散过程的随机正则化机制。

Flux through a time-periodic gate: Monte Carlo test of a homogenization result

We investigate via Monte Carlo numerical simulations and theoretical considerations the outflux of random walkers moving in an interval bounded by an interface exhibiting channels (pores, doors) which undergo an open/close cycle according to a periodic schedule. We examine the onset of a limiting boundary behavior characterized by a constant ratio between the outflux and the local density, in the thermodynamic limit. We compare such a limit with the predictions of a theoretical model already obtained in the literature as the homogenization limit of a suitable diffusion problem.

Periodic homogenization with an interface: The multi-dimensional case

We consider a diffusion process with coefficients that are periodic outside of an "interface region" of finite thickness. The question investigated in this article is the limiting long time/large scale behavior of such a process under diffusive rescaling. It is clear that outside of the interface, the limiting process must behave like Brownian motion, with diffusion matrices given by the standard theory of homogenization. The interesting behavior therefore occurs on the interface. Our main result is that the limiting process is a semimartingale whose bounded variation part is proportional to t…