Crossing dependence in Gaussian processes in time, space, and space-time - some applications of a generalized Rice's formula

The distribution of zeros of a Gaussian process was one of the unsolved problems in Steve Rice work, Mathematical analysis of random noise, 1944-45. Rice's formula gives the expected number, and higher moments have been used to bound the distribution function for the distance between two successive crossings. A much harder problem is the joint distribution of several crossing events, like the bivariate distribution of two successive crossing intervals. A similar question in oceanography concerns the relation between the observed period of an ocean wave, registered as the time between an upcrossing and the next downcrossing of the mean water level, and the physical size of the approaching wave. In the talk I will present exact formulas for these and similar distributions, and use accurate numerical tools for computation.