The exotic Laplacians generate the Markov processes given by distribution derivatives of white noise
The exotic Laplacians generate the Markov processes given by distribution derivatives of white noise
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奇异的拉普拉斯算子生成由白噪声的分布导数给出的马尔可夫过程
DOI：
10.1142/s0219025713500203
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Kimiaki Saito
中科院分区：
文献类型：
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作者：
Luigi Accardi;Un Cig Ji;Kimiaki Saito
对于每个a∈ℝ+，我们引入了与白噪声的a阶导数有关的布朗运动。我们证明了这个马尔可夫过程的生成元是2a阶奇异拉普拉斯函数，它是由一个合适的Hilbert空间(2a阶Cesáro空间)的一个标准正交基的元素的2a阶导数的Cesáro平均给出的。特别是，对于a=1/2，人们发现了通常的Lévy拉普拉斯函数，但在这种情况下，与白噪声的1/2导数的联系也是新的。用来实现这些目标的主要技术工具是由于Accardi和Smolyanov5将著名的塞萨罗定理推广到高阶算术平均而得到的一个结果的推广。这些估计和其他估计允许证明与任何≥1/2阶奇异拉普拉斯算子相关的热半群的存在性，并给出其无穷维富里叶变换的显式表达式。
We introduce, for each a ∈ ℝ+, the Brownian motion associated to the distribution derivative of order a of white noise. We prove that the generator of this Markov process is theexotic Laplacian of order 2a, given by the Cesàro mean of order 2a of the second derivatives along the elements of an orthonormal basis of a suitable Hilbert space (the Cesàro space of order 2a). In particular, for a = 1/2 one finds the usual Lévy Laplacian, but also in this case the connection with the 1/2-derivative of white noise is new. The main technical tool, used to achieve these goals, is a generalization of a result due to Accardi and Smolyanov5extending the well-known Cesàro theorem to higher order arithmetic means. These and other estimates allow to prove existence of the heat semi-group associated to any exotic Laplacian of order ≥ 1/2 and to give its explicit expression in terms of infinite dimensional Fourier transform.