Title: Persistence via exact excursion time distributions
Authors: Krzysztof Podgorski - Lund University (Sweden) [presenting]
Georg Lindgren - Lund University (Sweden)
Igor Rychlik - Chalmers University (Sweden)
Abstract: Finding the probability that a stochastic system stays in a certain region of its state space over a specified time -- a long-standing problem in computational physics, applied and theoretical mathematics -- is approached through the extended and multivariate Rice formula. In principle, it applies to any smooth multivariate process given that efficient numerical implementations of the high-dimensional integration are available. For Gaussian processes, the computations are effective and more precise than those based on the Rice series expansions and other approximations. It is shown that the approach can yield the explicit integral forms of a variety of excursion time distributional problems. It solves the two-step excursion dependence for a general stationary differentiable Gaussian process, in both theoretical and practical numerical sense. The solution is based on exact expressions for the probability density for one and two successive excursion lengths. The numerical routine RIND computes the densities using recent advances in scientific computing and is easily accessible for a general covariance function. Some analytical results are also offered that explain the effectiveness of the implemented method and points out how it can be utilized for non-Gaussian processes. The approach compares favorably with other methods. In particular, the approximation error is more controllable than in the independent and Markov interval approximations.