Dirichlet Forms Methods for Poisson Point Measures and Lévy Processes: With Emphasis on the Creation-Annihilation Techniques
Dirichlet Forms Methods for Poisson Point Measures and Lévy Processes: With Emphasis on the Creation-Annihilation Techniques
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DOI：
10.1007/978-3-319-25820-1
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发表时间：
2015-12
期刊：
影响因子：
--
通讯作者：
N. Bouleau;L. Denis
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文献类型：
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作者：
N. Bouleau;L. Denis
在随机分析领域，狄利克雷形式特别受到那些满足其惊人性质的人的喜爱。在将有限维势理论开放给希尔伯特技术之后，他们已经证明在无限维中对于维纳空间和扩散过程的研究以及对于适合于随机分析的误差演算非常有用，同时保留了一些神秘性--必须说--因为它们保留了一些令人惊讶的特征，如能量图像密度特性，这仍然是一个非常迷人的谜这本书的目的是实现他们的财产，以建立新的技术有关泊松措施和莱维过程是中央的研究过程与跳跃。更准确地说，我们开发了一种新的方法Malliavin演算泊松措施和跳跃过程。该方法产生了一个新的显式演算：如果一个认为泊松测度作为一个随机系统的粒子，它包括在添加一个粒子，并采取它回来后，计算梯度即“衍生”相对于位置的粒子（大小的跳跃的情况下，一个列维过程）。这种方法允许建立泊松泛函的绝对连续性，如Lévy区域，泊松测度驱动的解和许多其他例子。通过迭代梯度，我们还构造了相应的任意阶Sobolev空间，并得到了保证Poisson泛函律光滑性的一些判据。许多数学家的工作已经把我们的道路上这样一个计划，特别是保罗Malliavin，正俊福岛，塞尔吉奥Albeverio，迈克尔Roeckner和特别是弗朗西斯赫希我们特别感谢。我们的想法在几所大学的各种讲座和会议上逐渐形成了最终的形式，在那里进行了富有成效的讨论：在日本感谢Iroshi Sugita，Yasushi石川和Arturo Kohatsu-Higa，在威尔士斯旺西感谢Niels Jacob，尤金Lytvynov，2007年9月至10月在斯德哥尔摩的Mittag-Leffler研究所感谢玛尔塔Sanz，2009年6月在香港，感谢Nicolas Privault; 2012年在Roscoff，感谢Rainer Buckdahn。我们感谢这些同事。频繁
In the landscape of stochastic analysis, Dirichlet forms are particularly endearing to those who have met their amazing properties. After having opened the finite dimensional potential theory to Hilbertian techniques, they have proved very useful in infinite dimensions for the study of Wiener space and diffusion processes as well as for an error calculus adapted to stochastic analysis, while retaining some mystery––it must be said––due to the fact that they have kept several astonishing conjectures such as the energy image density property which remains a quite fascinating enigma to this day.The purpose of this book is to implement their properties to build new techniques concerning Poisson measures and Lévy processes that are central to the study of processes with jumps. More precisely, we develop a new approach to Malliavin calculus for Poisson measures and jump processes. The method gives rise to a new explicit calculus: if one considers a Poisson measure as a random system of particles, it consists in adding a particle and taking it back after computing the gradient ie “derivating” with respect to the position of the particle (the size of the jumps in the case of a Lévy process). This method permits to establish absolute continuity of Poisson functionals such as Lévy areas, solutions of SDE’s driven by Poisson measure and many other examples. By iterating the gradient we also construct associated Sobolev spaces of any order and get some criteria ensuring smoothness of the law of Poisson functionals. The work of many mathematicians have put us on the path of such a program, especially that of Paul Malliavin, Masatoshi Fukushima, Sergio Albeverio, Michael Roeckner and particularly Francis Hirsch to whom we are particularly acknowledging. Our ideas gradually took their final form during various lectures and conferences in several universities where fruitful discussions have been possible: in Japan thanks to Iroshi Sugita, Yasushi Ishikawa and Arturo Kohatsu-Higa, at Swansea Wales thanks to Niels Jacob, Eugene Lytvynov, in the Mittag-Leffler Institute Stockholm in September–October 2007 thanks to Marta Sanz, in Hong Kong in June 2009 thanks to Nicolas Privault and at Roscoff in 2012 thanks to Rainer Buckdahn. We acknowledge these colleagues gratefully. Frequent