Small deviation and geometric quantification
0405855 Gao The PI will study small deviation probabilities for Gaussian processes. In particular, he will look at three specific problems. First, he will study the relationship between small deviations under different norms. This can allow results obtained under one norm to be modified so that they are true under a different norm. Second, he will look at the small deviation probabilities for Gaussian processes with specific forms of the covariance kernel. Finally, he will study the small deviation asymptotic behavior of some Gaussian random fields including the Brownian sheet. The PI believes that some combination of the Karhunen-Loeve expansion, Fourier analysis and geometric methods will allow significant progress to be made on each of these problems. Stochastic processes move randomly through space and time. Simple descriptions of these processes tell where they should be located as a function of time (the mean process) and what their range should be. Probabilists have studied large deviations (events where the process is far away from where it should be) for many years. Now they are turning to small deviations (events where the range of the process is much smaller than it should be). This research focuses on small deviations for Gaussian processes (processes whose underlying structure is the normal distribution).