×

Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields. (English) Zbl 1165.60022

Summary: Let \(X = \{X(t), t\in \mathbb R^N\}\) be a Gaussian random field with values in \(\mathbb R^d\) defined by \(X(t) = (X_{1}(t), \dots , X_d(t))\), where \(X_{1}, \dots , X_d\) are independent copies of a centered Gaussian random field \(X_{0}\). Under certain general conditions on \(X_{0}\), we study the hitting probabilities of \(X\) and determine the Hausdorff dimension of the inverse image \(X^{-1}(F)\), where \(F {\subseteq} \mathbb R^d\) is a non-random Borel set. The class of Gaussian random fields that satisfy our conditions includes not only fractional Brownian motion and the Brownian sheet, but also such anisotropic fields as fractional Brownian sheets, solutions to stochastic heat equation driven by space-time white noise and the operator-scaling Gaussian random fields with stationary increments constructed by H. Biermé, M. M. Meerschaert and H.-P. Scheffler [Stochastic Process. Appl. 117, 312–332 (2007; Zbl 1111.60033)].

MSC:

60G60 Random fields
60G15 Gaussian processes
60G17 Sample path properties
28A80 Fractals

Citations:

Zbl 1111.60033
PDFBibTeX XMLCite
Full Text: DOI