Homogenization of convolution type semigroups in high contrast media

IF 1.4 3区 数学 Q1 MATHEMATICS
Andrey Piatnitski, Elena Zhizhina
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引用次数: 0

Abstract

The paper deals with the asymptotic properties of semigroups associated with Markov jump processes in a high contrast periodic medium in \(\mathbb {R}^d\), \(d\ge 1\). In order to study the limit behaviour of these semigroups we equip the corresponding Markov processes with an extra variable that characterizes the position of the process inside the period, and show that the limit dynamics of these two-component processes remains Markov. We describe the limit process and prove the convergence of the corresponding semigroups as well as the convergence in law of the extended processes in the path space. Since the components of the limit process are coupled, the dynamics of the first (spacial) component need not have a semigroup property. We derive the evolution equation with a memory term for the dynamics of this component of the limit process. We also discuss the construction of the limit semigroup in the \(L^2\) space and study the spectrum of its generator.

高对比度介质中卷积型半群的均匀化
研究了高反差周期介质中马尔可夫跳变过程半群的渐近性质 \(\mathbb {R}^d\), \(d\ge 1\)。为了研究这些半群的极限行为,我们给相应的马尔可夫过程赋予一个额外的变量来表征过程在周期内的位置,并证明了这些双组分过程的极限动力学仍然是马尔可夫的。我们描述了极限过程,并证明了相应半群的收敛性以及扩展过程在路径空间上的收敛性。由于极限过程的分量是耦合的,所以第一个(空间)分量的动力学不必具有半群性质。我们导出了极限过程中这一分量的动力学的带记忆项的演化方程。文中还讨论了一类极限半群的构造 \(L^2\) 空间和研究其发生器的频谱。
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来源期刊
Analysis and Mathematical Physics
Analysis and Mathematical Physics MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
2.70
自引率
0.00%
发文量
122
期刊介绍: Analysis and Mathematical Physics (AMP) publishes current research results as well as selected high-quality survey articles in real, complex, harmonic; and geometric analysis originating and or having applications in mathematical physics. The journal promotes dialog among specialists in these areas.
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