Real Self-Similar Processes Started from the Origin
Real Self-Similar Processes Started from the Origin
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DOI：
10.1214/16-aop1105
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发表时间：
2015-01
期刊：
影响因子：
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通讯作者：
S. Dereich;L. Doering;A. Kyprianou
中科院分区：
文献类型：
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作者：
S. Dereich;L. Doering;A. Kyprianou
自从Lamperti的开创性工作以来，人们对理解自相似马尔可夫过程的一般结构产生了很大的兴趣。Lamperti给出了初始条件严格大于0的正自相似马尔可夫过程的表示，并将其推广到零初始条件。对于实自相似马尔可夫过程（rssMps），推广了Lamperti表示，给出了初始条件不同于原点的马尔可夫加性过程与实自相似马尔可夫过程的一一对应关系。我们发展了马尔可夫加性过程的涨落理论，并利用库兹涅佐夫测度构造了从原点出发的暂态实自相似马尔可夫过程的定律。该构造通过双边马尔可夫加性过程给出了一个路径表示，将Lamperti-Kiu表示扩展到原点。
Since the seminal work of Lamperti there is a lot of interest in the understanding of the general structure of self-similar Markov processes. Lamperti gave a representation of positive self-similar Markov processes with initial condition strictly larger than 0 which subsequently was extended to zero initial condition. For real self-similar Markov processes (rssMps) there is a generalization of Lamperti's representation giving a one-to-one correspondence between Markov additive processes and rssMps with initial condition different from the origin. We develop fluctuation theory for Markov additive processes and use Kuznetsov measures to construct the law of transient real self-similar Markov processes issued from the origin. The construction gives a pathwise representation through two-sided Markov additive processes extending the Lamperti-Kiu representation to the origin.