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 (-T1,T),T>T1>>1 and the integrated FBM in(0,T), T>>1 .


The problem
be a real-valued stochastic process with the following asymptotics: where x  is the so-called survival exponent of ) (t x . Below we focus on estimating x  for some selfsimilar Gaussian processes in extended intervals Usually the estimation of the survival exponents is based on Slepian's lemma. To do this we need 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), ) (t w H , of order 1 0   H both with one-and multidimensional time. According to Molchan (1999)  , 0 ( T T   (Sinai, 1992) ( The nature of this result is best understood in terms of a series of generalizations where the integrand is random walk with discrete or continuous time (see, e.g., Isozaki and Watanabe, 1994;Isozaki and Kotani, 2000;Simon,2007;Vysotsky,2010;Aurzada and Dereich, 2011;Dembo and Gao, 2011). The extension of (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: with the same exponent x x    ( Molchan,1999( Molchan, ,2008. More generally, the dual exponent is defined by the asymptotics ))) 1 ( 1 ( exp( )) 1 , To formulate the simplest condition of the exponent equality, we define one more exponent x   by means of the asymptotics where * T t is the position of the maximum of ) (t x in T  , i.e., ) ),  Khokhlov (2003, 2004) Proof. The identity of dual exponents for ) (t I H follows from (Molchan and Khokhlov, 2004); the dual survival exponent exists, because the dual correlation function is positive .The inequality (a) is a consequence of the relation To prove (b, c) , we use the correlation function of the process ) ( implies (c) for all H . A test of the pure analytical facts (7, 9, and 10) is given in Appendix.
Remark 1.The proposition 1а follows from the more informative relation This inequality is important for understanding the numerical result by Molchan and Khokhlov(2003) represented in the form of the empirical estimates of H I ~ in Figure 1. We can see that the empirical esti-  Molchan and Khokhlov,2003).

Laplace transform of white noise.
Consider the process Proof . The exponent equality for the dual processes L and L follows from Lemma 1 for random polynomials with the standard Gaussian independent coefficients (Dembo et al., 2002). A continuous analogue of the polynomial on any of four intervals

Fractional Slepian's process.
We reserve this term for a Gaussian stationary process Lemma 2. (Li and Shao, 2004). Let ) (t x be a centered Gaussian stationary process with a finite non- Remark 3. Lemma 1 was derived by Li and Shao (2004) for the Slepian process , ) ( 2 / 1 t S , but the proof remains valid for the general case. There is an explicit but very complicated formula for (Shepp, 1971). In case of Remark 4. Lemma 3 is a version of Proposition 1.6 from the paper by Aurzada and Dereich (2011) where the left inequality holds for

Corollary: odd component of the fractional Brownian motion.
Its dual stationary process  H w has the following correlation function: Krug et al (1997) estimated the exponent as follows: For small H these estimates are one-sided only. The following inequality Heuristically this can be explained as follows. As  and are independent standard Gaussian variables. The probability (4) for the limiting processes are quite different: Unfortunately, this argument is insufficient to predict the behavior of H S  for small H , because the step  cannot be arbitrary and is a function of H .

Khanin's problem.
The survival exponent for fractional Brownian motion in the intervals This interesting fact follows from both self-similarity of H w and the stationarity of its increments (Molchan, 1999 was asked by K.Khanin. The next proposition contains a partial answer to this question.

Remark 6. To clarify why
pendent standard Gaussian variables .The probability (1) for the limit sequence is is a slowly varying function, whereas for the limit sequence of where ) (x  is the Gaussian distribution function. As in Remark 5, we have non-trivial exponential asymptotics where the threshold for } { k  is constant or bounded. Indeed, the event in (23) yields the ine-

Explicit value of
We have two explicit but isolated results for the fractional Brownian motion: These results can be combined as follows: It is easy to see that Let H m be the median of the random variable ) ), in Lemma 2 and using notation (17) (see Molchan and Khokhlov, 2004).

Estimation of
for any s t, , we have, by Slepian's lemma, Distribution of *  t . We remind the main properties of the distribution function, ) Molchan,1999;Molchan and Khokhlov,2004): By (26, 27), Using (26, 28), one has where 5 . 0   . We have used here the existence and continuity of ) ( and (30, 31) give a lower bound of T p : Here and below Using (29, 31), we get an upper bound on T p :

Relation (7):
By (6), one has for small and large t )) ( where H H   1 .Therefore, we have the following asymptotics for These relations support (7) both for small and large enough t. To verify (7) in the general case, we consider the following test function: We have to show that Obviously, ' f has a single zero in (0,1), i.e. f has a unique extreme point. But ) Numerical testing supports the desired inequality 0   for interior points of S .
Comment. Our preliminary numerical test was concerned with points of grid with step 0.005. The first derivatives of are uniformly bounded from above on S . This fact helps to find a final grid step to prove 0   for all interior points of S . The corresponding analysis is unwieldy and so is omitted.

Relation (9):
, we will have the following representation for the test function: in a neighborhood f two sides of S: x=0 and x=1, because The same is true for other sides: is a concave function with two zeroes in (0,1), because has unique zero in (0,1) and ) (x a  has unique extremum.
So, we prove that  and the relation between p and  : . The numerical testing supports this conclusion for interior of S (see more in the Comment from the Appendix section 'Relation 7').

Relations
Taking into account the asymptotics of  near 0, we come to a necessary condition for to be negative, It is easy to see, that both terms in parentheses are positive on (0.5, 1).

Figure 1
The survival exponents