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Zbl 0894.60036
Roux, Daniel
Multi-scale analysis of Gaussian Markovian processes near a singular point. (Analyse multi-échelle d'un processus gaussien markovien au voisinage d'une singularité.) (French)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 33, No.3, 295-322 (1997). ISSN 0246-0203

Summary: Let $X$ be some Gaussian process. We suppose $X$ to be Markovian of order one, which means the topology of its reproducing kernel Hilbert space $H_X$ is given by a symmetric Dirichlet form ${\cal A}$ of order two. A point $y$ is said to be singular when the leading coefficient of the form ${\cal A}$ vanishes or becomes infinite. We analyze the regularity of the trajectories of such a process $X$ near any isolated singular point $y$. In order to do this, $X$ is decomposed on a wavelet basis of $H_X$. The behavior of $X$ is given after a precise study of the wavelets near $y$. Using the same ideas the form ${\cal A}$ can also be identified in the vicinity of a singular point.

MSC 2000:: *60G15 Gaussian processes

Keywords: Gaussian process; symmetric Dirichlet form; regularity of the trajectories; wavelet basis; wavelets

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