Survival Exponents for Some Gaussian Processes

The problem is a power-law asymptotics of the probability that a self-similar process does not exceed a fixed level during long time. The exponent in such asymptotics is estimated for some Gaussian processes, including the fractional Brownian motion (FBM) in , and the integrated FBM in , . 1. The Problem Let be a real-valued stochastic process with the following asymptotics: where is the so-called survival exponent of . Below we focus on estimating for some self-similar Gaussian processes in extended intervals and , . Usually the estimation of the survival exponents is based on Slepian’s lemma. The estimation requires reference processes with explicit or almost explicit values of . Unfortunately, the list of such processes is very short. This includes the fractional Brownian motion (FBM), , of order both with one- and multidimensional time. According to Molchan ([1]) Another important example is the integrated Brownian motion with the exponent (Sinai [2]). The nature of this result is best understood in terms of a series of generalizations where the integrand is a random walk with discrete or continuous time (see, e.g., Isozaki and Watanabe [3]; Isozaki and Kotani [4]; Simon [5]; Vysotsky [6, 7]; Aurzada and Dereich [8]; Dembo et al. [9]; Denisov and Wachtel [10]. The extension of (1.3) to include the case of the integrated fractional Brownian motion, , remains an important; but as yet unsolved problem. Below we consider the survival exponents for the following Gaussian processes: ; FBM in , ; the Laplace transform of white noise with ; the fractional Slepian’s stationary process whose correlation function is , . Our approach to the estimation of is more or less traditional. Namely, any self-similar process in generates a dual stationary process , , where is the self-similarity index of . For a large class of Gaussian processes, relation (1.1) induces the dual asymptotics with the same exponent , [1, 11]. More generally, the dual exponent is defined by the asymptotics To formulate the simplest condition for the exponents to be equal, we define one more exponent by means of the asymptotics where is the position of the maximum of in , that is, . Lemma 1.1 (see [1, 11]). Let be a self-similar continuous Gaussian process in and be the reproducing kernel Hilbert space associated with . Suppose that there exists such an element of that and . Then , and can exist simultaneously only; moreover, the exponents are equal to each other. The equality reduces the original problem to the estimation of . Nonnegativity of the correlation function of guarantees the