We consider the one-sided exit problem for (fractionally) integrated random walks and L 'evy processes. We prove that the rate of decrease of the non-exit probability -- the so-called survival exponent -- is universal in this class of processes. In particular, the survival exponent can be inferred from the (fractionally) integrated Brownian motion. This, in particular, extends Sinai's result on the survival exponent for the integrated simple random walk to general random walks with some finite exponential moment. Further, we prove existence and monotonicity of the survival exponent of fractionally integrated processes. We show that this exponent is related to a constant appearing in the study of random polynomials.
The survival exponent is unknown for the integrated fractional Brownian motion, see @cite_34 . A related question for the Brownian sheet is solved in @cite_7 @cite_5 . Further references with partial results are @cite_29 @cite_1 @cite_19 .
@cite_7 @cite_29 @cite_1 @cite_19 @cite_5 @cite_34
2128129237 2090049826 2040360177 1993598763
Let ( = _x ;x R ) be a mean—zero Gaussian process with covariance @math Here @math where ψ is a non—negative even function on R. Note that o is an even function and o(0) = 0. In this paper we assume that Let X be a symmetric stable process of index α∈ (1,2] and let L x t denote the local time at time t and position x. Let V(t) be such that L t V(t) = sup x∈ ℝL t x . We call V(t) the most visited site of X up to time t. We prove the transience of V, that is, lim t →∞ |V(t)| = ∞ almost surely. An estimate is given concerning the rate of escape of V. The result extends a well-known theorem of Bass and Griffin for Brownian motion. Our approach is based upon an extension of the Ray–Knight theorem for symmetric Markov processes, and relates stable local times to fractional Brownian motion and further to the winding problem for planar Brownian motion. Let @math denote the first passage time to 1 of a standard Brownian motion. It is well known that as @math goes to infinity, @math goes to zero at rate @math , where @math equals @math . The goal of this note is to establish a quantitative, infinite dimensional version of this result. Namely, we will prove the existence of positive and finite constants @math and @math , such that for all @math , @math where @math ' denotes the natural logarithm, and @math is the Fukushima-Malliavin capacity on the space of continuous functions. We consider the integral of fractional Brownian motion (IFBM) and its functionals ξ T on the intervals (0,T) and (−T,T) of the following types: the maximum M T , the position of the maximum, the occupation time above zero etc. We show how the asymptotics of P(ξ T <1)=p T ,T→∞, is related to the Hausdorff dimension of Lagrangian regular points for the inviscid Burgers equation with FBM initial velocity. We produce computational evidence in favor of a power asymptotics for p T . The data do not reject the hypothesis that the exponent θ of the power law is related to the similarity parameter H of fractional Brownian motion as follows: θ=−(1−H) for the interval (−T,T) and θ=−H(1−H) for (0,T). The point 0 is special in that IFBM and its derivative both vanish there.