# Survival Exponents for Some Gaussian Processes

Authors: G. Molchan
Journal: International Journal of Stochastic Analysis
 (2012)

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## Abstract

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(−T−,T), T≥T−≫1 and the integrated FBM in (0,T), T≫1.


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## Body

## 1. The Problem

Letx(t),x(0)=0 be a real-valued stochastic process with the following asymptotics:
(1.1)P(x(t)<1,t∈ΔT)=T-θx+o(1),T→∞,
where θx is the so-called survivalexponent of x(t). Below we focus on estimating θx for some self-similar Gaussian processes in extended intervals ΔT=(0,T) and (-T-,T), T≥T-≫1. 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 θ. Unfortunately, the list of such processes is very short. This includes the fractional Brownian motion (FBM), wH(t), of order 0<H<1 both with one- and multidimensional time. According to Molchan ([1])
(1.2)θwH=1-HforΔT=(0,T),θwH=dforΔT=(-T,T)d.
Another important example is the integrated Brownian motion I(t)=∫0tw(s)ds with the exponent
(1.3)θI=14,ΔT=(0,T),
(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, IH(t)=∫0twH(s)ds, remains an important; but as yet unsolved problem.Below we consider the survival exponents for the following Gaussian processes:IH(t),t∈(0,T);χH(t)=sign(t)wH(t),t∈(-T,T); FBM in ΔT=(-Tα,T), 0≤α≤1; the Laplace transform of white noise with ΔT=(0,T); the fractional Slepian’s stationary process whose correlation function is BSH(t)=(1-|t|2H)+, 0<H≤1/2.Our approach to the estimation ofθ is more or less traditional. Namely, any self-similar process x(t) in ΔT=(0,T) generates a dual stationary processx~(s)=e-hsx(es), s<lnT:=T~, where h is the self-similarity index of x(t). For a large class of Gaussian processes, relation (1.1) induces the dual asymptotics
(1.4)P(x~(s)≤0,0<s<T~)=exp(-θ~xT~(1+o(1))),T~→∞,
with the same exponent θ~x=θx, [1, 11]. More generally, the dual exponent is defined by the asymptotics
(1.5)P(x(t)≤0,t∈ΔT∖(-1,1))=exp(-θ~xT~(1+o(1))).
To formulate the simplest condition for the exponents to be equal, we define one more exponent θ˘x by means of the asymptotics
(1.6)P(|tT*|≤1)=T-θ˘x+o(1),
where tT* is the position of the maximum of x(t) in ΔT, that is, x(tT*)=sup(x(t),t∈ΔT).Lemma 1.1 (see [1, 11]).
Letx(t),x(0)=0 be a self-similar continuous Gaussian process in ΔT=(-T-,T),T-≤T and (Hx(ΔT),∥∥T) be the reproducing kernel Hilbert space associated with x(t). Suppose that there exists such an element φ of Hx(ΔT) that φ(t)≥1,|t|>1 and ∥φ∥T2=o(lnT). Then θx,θ~x, and θ˘x can exist simultaneously only; moreover, the exponents are equal to each other.The equalityθ=θ~ reduces the original problem to the estimation of θ~. Nonnegativity of the correlation function of x~(s) guarantees the existence of the exponent θ~, [12]. In turn, the inequality of two correlation functions, B1(s)≤B2(s), Bi(0)=1, implies, by Slepian’s lemma, the inverted inequality for the corresponding exponents: θ~1≥θ~2.An essentially different approach is required to find the explicit value ofθ for FBM in ΔT=(-Tα,T) and to estimate θ~ in (1.4) for the fractional Slepian process with a small parameter H.


## 2. Examples

### 2.1. Integrated Fractional Brownian Motion

Consider the process(2.1)IH(t)=∫0twH(s)ds,
where wH(t) is the fractional Brownian motion, that is, a Gaussian random process with the stationary increments: E|wH(t)-wH(s)|2=|t-s|2H, wH(0)=0. Molchan and Khokhlov [13, 14] analyzed the exponent θIH theoretically and numerically and formulated the following Hypothesis: θIH=H(1-H) for ΔT=(0,T)) and θIH=1-H for ΔT=(-T,T).The unexpected symmetryθIH=θI1-H for ΔT=(0,T) caused some doubt as to the numerical results. To support the hypothesis, Molchan [11] derived the following estimates of θ for IH(t):
(2.2)ρH(1-H)≤θIH+≤θIH-/+≤(1-H),
where ρ is a small constant and (+) and (-/+) are indicators of the intervals ΔT=(0,T) and ΔT=(-T,T), respectively. Note that, in the case of H<1/2 and ΔT=(-T,T), it is unknown whether the exponent exists. In such cases we have to operate with upper θ¯ and lower θ_ exponents. Therefore, θIH-/+ in (2.2) for H<1/2 is any number from the interval (θ_,θ)¯. The relation (2.2) can be improved as follows.Proposition 2.1.
ForΔT=(0,T), one has(a)
θIH≥θI1-H,0<H≤0.5,(b)
0.5(H∧H-)≤θIH≤H-,H-=1-H,(c)
θIH≤(1-(H∧H-)2)/12.Proof.
The identity of the dual exponents forIH(t) follows from [14]; the dual survival exponent exists because the dual correlation function,
(2.3)B~IH(s)=(2+4H)-1[(2+2H)(eHs+e-Hs)-e(1+H)s-e-(1+H)s+(es/2-e-s/2)2H+2]
is positive. The inequality (a) is a consequence of the relation
(2.4)B~IH(t)≤B~I1-H(t),0<H≤12.
To prove (b, c), we use the correlation function of the processI~1/2(ps), that is,
(2.5)B~I1/2(ps)=12(3exp(-p|s|2)-exp(-3p|s|2)),
and the respective exponent θ~=p/4 (see (1.3)). The relation
(2.6)B~IH(t)≤B~I1/2(pt),H≥12,p=2(1-H),
implies θIH≥(1-H)/2 for H≥1/2. Using (a) in addition, we come to the lower bound in (b) because θIH≥θI1-H≥H/2 for H≤1/2.
Similarly, the relation(2.7)B~IH(t)≥B~I1/2(pt),H≤12,p=21-H23,
implies (c) for all H. A test of the purely analytical facts (2.4), (2.6), and (2.7) is given in the appendix.Remark 2.2.
Proposition2.1(a) follows from the more informative relation:
(2.8)P(I~H(s)≤0,s∈(0,T~))≤P(I~1-H(s)≤0,s∈(0,T~)).
This inequality is important for understanding the numerical result by Molchan and Khokhlov [13] represented in the form of empirical estimates of θ~IH in Figure 1. We can see that the empirical estimates show small but one-sided deviations from the hypothetical curve θ=H(1-H) before and after H=1/2. The signs of these deviations are consistent with (2.8), while the amplitudes are compatible with the model
(2.9)P(I~H(s)≤0,s∈(0,T~))≈CT~α(H)exp(-H(1-H)T~),T~≫1,sgnα(H)=sign(H-0.5),
and α(H)=H-0.5 (more can be found in [13]).Figure 1
The survival exponentsθ~IH for the integrated fractional Brownian motion in ΔT=(-T,T): hypothetical values (parabolic line), empirical estimates (small circles, squares), and theoretical bounds (shaded zonegiven by Proposition 2.1(b, c)). The empirical exponents are based on the model ((2.9), α(H)=0) in three time intervals of T~=lnT:ln(1/ε)≤T~(1-H)H≤ln(10/ε), where ε=0.01, 0.003, and 0.001 (see more in [13]).


### 2.2. The Laplace Transform of White Noise

Consider the processL(t)=t∫0∞e-tudw(u), where w(u) is Brownian motion. The dual stationary process L~(s) has the correlation function B~L(s)=1/cosh(s/2). Using (2.5) as a majorant of B~L(s), we improve the lower bound of θ~L as follows.Proposition 2.3.
3
-
1
/
2
≤
4
θ
~
L
≤
1.Proof.
That the exponents for the dual processesL and L~ are equal follows from Lemma 1.1 with φ(t)=t(1+εT)/(t+εT), where εT=1/lnT. For indeed, φ(t)=EL(t)η, where η=(1+εT-1)L(εT). By definition of the Hilbert space HL(ΔT), we have the desired estimate:
(2.10)∥φ∥T2=Eη2=εT-1(εT+1)22=O(lnT).
By (1.3) and Slepian’s lemma, the relation
(2.11)B~I1/2(t)≤B~L(pt),p≤1
has as a consequence the estimate 4pθ~L≤1. The opposite inequality
(2.12)B~I1/2(t)≥B~L(pt),p2≥3,
implies 4pθ~L≥1. The test of ((2.11), p=1) and ((2.12), p=2) is very simple and yields the Li and Shao [12, 15] estimates: 0.5<4θ~L<1. The appendix contains a proof of (2.11), (2.12) for all interesting values of p: 1,2, and 3.Remark 2.4.
Thedual survival exponent of L(t) is of interest as a parameter of the following asymptotic relation:
(2.13)P(∑02nξixi≠0,x∈R1)=(2n)-4θ~L+o(1),n→∞,
for random polynomials with the standard Gaussian independent coefficients [16]. A continuous analogue of the polynomial on any of four intervals 0<±x±1≤1 is the Laplace transform of white noise, which partially explains the appearance of θ~L in the asymptotic relation (2.13). Simulations suggest 4θ~L=0.76±0.03, [16] and 4θ~L≈0.75, [17].


### 2.3. Fractional Slepian’s Process

We reserve this term for a Gaussian stationary processSH(t) with correlation function
(2.14)BSH(t)=(1-|t|2H)+,0<H≤12,
because S1/2(t) is known as the Slepian process and SH(t)-SH(0),0<t≤1, is equal in distribution to the fractional Brownian motion on the interval (0,1). By the Polya criterion, the fractional Slepian process exists because BSH(t) is a nonincreasing and a convex function on the semiaxis. The fact of the correlation function being nonnegative guarantees the existence of θ~SH in (1.4). SH(t) can be useful as a reference process in estimation of the survival exponents. Therefore it is important to have accurate estimates of the exponent for SH(t). The case of small H is the most interesting because it describes a transition of SH(t) to white noise. Our estimates of θ~SH are based on two lemmas, where we use the following notation:
(2.15)θ~(f,Δ)=-|Δ|-1logP(x(t)≤f(t),t∈Δ).Lemma 2.5 (see [12]).
Letx(t) be a centered Gaussian stationary process with a finite nonnegative correlation function, that is, Bx(t)≥0 and Bx(t)=0 for |t|≥T0. Then the limit
(2.16)θ~(a)=limT→∞θ~(a,(0,T)),
exists for every a∈R1. Moreover,
(2.17)(1+1k)-1θ~(a,kΔ0)≤θ~(a)≤θ~(a,kΔ0),Δ0=(0,T0).Remark 2.6.
Lemma1.1 was derived by Li and Shao [12] for the Slepian process, S1/2(t), but the proof remains valid for the general case. There is an explicit but very complicated formula for θ~SH(0,Δ) with H=1/2 [18]. In case of Δ=(0,2), this result reduces to
(2.18)P(S1/2(t)≤0,t∈(0,2))=16-2+38π
and gives 1.336<θ~S1/2<2.004.Lemma 2.7 (see [8]).
Letx(t) be a centered Gaussian process in an interval Δ with a correlation function B(t,s) and (Hx(Δ),∥·∥Δ) be the Hilbert space with the reproducing kernel B(t,s) on Δ×Δ. If 0<θ~(a,Δ)<∞, then
(2.19)|θ~(a+f,Δ)-θ~(a,Δ)|≤∥f∥Δ2|Δ|.Remark 2.8.
Lemma2.7 is a version of Proposition  1.6 from the paper by Aurzada and Dereich [8]; relation (2.19) successfully supplements the original Lemma 1.1.Proposition 2.9.
The persistence exponent of processSH(t) has the following estimates:
(2.20)-(1-H)H-1ln(2H)≤θ~SH≤49H-2,
where the left inequality holds for 0<H≤e-2/2.Corollary 2.10.
IfwH-(t)=(wH(t)-wH(-t))/2 is the odd component of the fractional Brownian motion, then
(2.21)θ~wH-≤(7/H)22,0<H<0.5.Proof.
The dual stationary processw~H- has the correlation function
(2.22)B~wH-(t)=(cosht2)2H-(sinht2)2H,
which is positive. Hence the exponent θ~wH- exists. The inequality
(2.23)B~wH-(2t)=(cosht)2H(1-(tanht)2H)≥(cosht)2H(1-|t|2H)+≥B~SH(t),
and Proposition 2.9 immediately imply the corollary.Remark 2.11.
The following estimates ofθ~wH- are due to Krug et al. [19]:
(2.24)θ~wH-≥min((1-H)2H,(1-H)21/(2H)-1),0<H<0.5,θ~wH-≤(1-H)2H,0.1549<H<0.5.
For small H these estimates are one-sided only.Remark 2.12.
A considerable difference in the behavior ofθ~wH- and θ~wH=1-H for small H is expected. Heuristically this can be explained as follows. As H→0, the discrete processes w~H-(kΔ) and w~H(kΔ) have different weak limits: {ξk} and {ξk-η/2}, respectively, where {ξk} and η are independent standard Gaussian variables. The probability (1.4) for the limiting processes is quite different:(2.25)P(ξk<0,k=1÷N)=2-N,P(ξk-η≤0,k=1÷N)=(N+1)-1.
Unfortunately, this argument fails to predict the behavior of θ~SH for small H, because the step Δ cannot be arbitrary and is a function of H.


### 2.4. Khanin’s Problem

The survival exponent for fractional Brownian motion in the intervalsΔT=(-T,T) is independent of the parameter H: θwH=1. This interesting fact follows from both self-similarity of wH and the stationarity of its increments [1].In the caseH<0.5, the variables wH(t) and wH(-t) are positive correlated. Therefore, a possible power-law asymptotics
(2.26)P(wH(t)<1,-wH(-t)<1,t∈(0,T))=T-θ+o(1),
where we change sign before wH(t) for negative t only, may have a radically different exponent compared with θwH=1. The question of finding bounds on the exponent θχH for the process
(2.27)χH(t)=sign(t)wH(t),ΔT=(-T,T),
was asked by K. Khanin. The next proposition contains a partial answer to this question.Proposition 2.13.
(
1
)
In the case 0.5≤H<1, the exponent θχH for ΔT=(-T,T) exists and admits of the following estimates:
(2.28)1<θχH(1-H)-1≤2,0.5≤H<1,in addition,θχ1/2=1.(2)
Letθ_χH be the lower exponent in (2.26), then
(2.29)θ_χH(1-H)-1≥(H-1-1)∧21/2H-1,0<H<0.25,θ_χH(1-H)-1≥20.25<H≤0.5.Remark 2.14.
To clarify whyθχH/θwH is unbounded for small H in the case ΔT=(-T,T), we consider again the limiting sequence for wH(kΔ) as H→0. This is {(ξk-ξ0)/2}, where the {ξk} are independent standard Gaussian variables. The probability (1.1) for the limit sequence is
(2.30)P{ξk<ξ0+2,|k|≤N}=(2N+1)-1l(N),
where l(N) is a slowly varying function, whereas for the limit sequence of χH(kΔ) we have
(2.31)P{ξ-k-2<ξ0<ξk+2,0<k≤N}≈πeN-1/2Φ(2)2N,
where Φ(x) is the Gaussian distribution function. As in Remark 2.12, we have nontrivial exponential asymptotics where the threshold for {ξk} is constant or bounded. Indeed, if the event in (2.31) is true, then
(2.32)|ξ0|<2+max(|∑1Nξ-k|,|∑1Nξk|)N=2+Op(1)N.


### 2.5. An Explicit Value ofθx

We have two explicit but isolated results for the fractional Brownian motion:θwH=(1-H) for ΔT=(0,T) and θwH=1 for ΔT=(-T,T). These results can be combined as follows.Proposition 2.15.
IfΔT=(-Tα,T), 0≤α≤1, then θwH=αH+(1-H).Remark 2.16.
The result is based on the following properties of the positiontΔ* of the maximum of wH(t) in Δ=[0,1]: tΔ* has a continuous probability density fΔ*(t) in (0, 1) and fΔ*(t)≈O(t-H) as t→0. In the case of multidimensional time, the behavior of fΔ*(t),Δ=(0,1)d near t=0 is a key to the survival exponent of wH(t) for ΔT=(-Tα,T)d,0<α<1 and H<1. By (1.2), θwH=d in the case α=1, and θwH=αd in the degenerate case: H=1.


## 2.1. Integrated Fractional Brownian Motion

Consider the process(2.1)IH(t)=∫0twH(s)ds,
where wH(t) is the fractional Brownian motion, that is, a Gaussian random process with the stationary increments: E|wH(t)-wH(s)|2=|t-s|2H, wH(0)=0. Molchan and Khokhlov [13, 14] analyzed the exponent θIH theoretically and numerically and formulated the following Hypothesis: θIH=H(1-H) for ΔT=(0,T)) and θIH=1-H for ΔT=(-T,T).The unexpected symmetryθIH=θI1-H for ΔT=(0,T) caused some doubt as to the numerical results. To support the hypothesis, Molchan [11] derived the following estimates of θ for IH(t):
(2.2)ρH(1-H)≤θIH+≤θIH-/+≤(1-H),
where ρ is a small constant and (+) and (-/+) are indicators of the intervals ΔT=(0,T) and ΔT=(-T,T), respectively. Note that, in the case of H<1/2 and ΔT=(-T,T), it is unknown whether the exponent exists. In such cases we have to operate with upper θ¯ and lower θ_ exponents. Therefore, θIH-/+ in (2.2) for H<1/2 is any number from the interval (θ_,θ)¯. The relation (2.2) can be improved as follows.Proposition 2.1.
ForΔT=(0,T), one has(a)
θIH≥θI1-H,0<H≤0.5,(b)
0.5(H∧H-)≤θIH≤H-,H-=1-H,(c)
θIH≤(1-(H∧H-)2)/12.Proof.
The identity of the dual exponents forIH(t) follows from [14]; the dual survival exponent exists because the dual correlation function,
(2.3)B~IH(s)=(2+4H)-1[(2+2H)(eHs+e-Hs)-e(1+H)s-e-(1+H)s+(es/2-e-s/2)2H+2]
is positive. The inequality (a) is a consequence of the relation
(2.4)B~IH(t)≤B~I1-H(t),0<H≤12.
To prove (b, c), we use the correlation function of the processI~1/2(ps), that is,
(2.5)B~I1/2(ps)=12(3exp(-p|s|2)-exp(-3p|s|2)),
and the respective exponent θ~=p/4 (see (1.3)). The relation
(2.6)B~IH(t)≤B~I1/2(pt),H≥12,p=2(1-H),
implies θIH≥(1-H)/2 for H≥1/2. Using (a) in addition, we come to the lower bound in (b) because θIH≥θI1-H≥H/2 for H≤1/2.
Similarly, the relation(2.7)B~IH(t)≥B~I1/2(pt),H≤12,p=21-H23,
implies (c) for all H. A test of the purely analytical facts (2.4), (2.6), and (2.7) is given in the appendix.Remark 2.2.
Proposition2.1(a) follows from the more informative relation:
(2.8)P(I~H(s)≤0,s∈(0,T~))≤P(I~1-H(s)≤0,s∈(0,T~)).
This inequality is important for understanding the numerical result by Molchan and Khokhlov [13] represented in the form of empirical estimates of θ~IH in Figure 1. We can see that the empirical estimates show small but one-sided deviations from the hypothetical curve θ=H(1-H) before and after H=1/2. The signs of these deviations are consistent with (2.8), while the amplitudes are compatible with the model
(2.9)P(I~H(s)≤0,s∈(0,T~))≈CT~α(H)exp(-H(1-H)T~),T~≫1,sgnα(H)=sign(H-0.5),
and α(H)=H-0.5 (more can be found in [13]).Figure 1
The survival exponentsθ~IH for the integrated fractional Brownian motion in ΔT=(-T,T): hypothetical values (parabolic line), empirical estimates (small circles, squares), and theoretical bounds (shaded zonegiven by Proposition 2.1(b, c)). The empirical exponents are based on the model ((2.9), α(H)=0) in three time intervals of T~=lnT:ln(1/ε)≤T~(1-H)H≤ln(10/ε), where ε=0.01, 0.003, and 0.001 (see more in [13]).


## 2.2. The Laplace Transform of White Noise

Consider the processL(t)=t∫0∞e-tudw(u), where w(u) is Brownian motion. The dual stationary process L~(s) has the correlation function B~L(s)=1/cosh(s/2). Using (2.5) as a majorant of B~L(s), we improve the lower bound of θ~L as follows.Proposition 2.3.
3
-
1
/
2
≤
4
θ
~
L
≤
1.Proof.
That the exponents for the dual processesL and L~ are equal follows from Lemma 1.1 with φ(t)=t(1+εT)/(t+εT), where εT=1/lnT. For indeed, φ(t)=EL(t)η, where η=(1+εT-1)L(εT). By definition of the Hilbert space HL(ΔT), we have the desired estimate:
(2.10)∥φ∥T2=Eη2=εT-1(εT+1)22=O(lnT).
By (1.3) and Slepian’s lemma, the relation
(2.11)B~I1/2(t)≤B~L(pt),p≤1
has as a consequence the estimate 4pθ~L≤1. The opposite inequality
(2.12)B~I1/2(t)≥B~L(pt),p2≥3,
implies 4pθ~L≥1. The test of ((2.11), p=1) and ((2.12), p=2) is very simple and yields the Li and Shao [12, 15] estimates: 0.5<4θ~L<1. The appendix contains a proof of (2.11), (2.12) for all interesting values of p: 1,2, and 3.Remark 2.4.
Thedual survival exponent of L(t) is of interest as a parameter of the following asymptotic relation:
(2.13)P(∑02nξixi≠0,x∈R1)=(2n)-4θ~L+o(1),n→∞,
for random polynomials with the standard Gaussian independent coefficients [16]. A continuous analogue of the polynomial on any of four intervals 0<±x±1≤1 is the Laplace transform of white noise, which partially explains the appearance of θ~L in the asymptotic relation (2.13). Simulations suggest 4θ~L=0.76±0.03, [16] and 4θ~L≈0.75, [17].


## 2.3. Fractional Slepian’s Process

We reserve this term for a Gaussian stationary processSH(t) with correlation function
(2.14)BSH(t)=(1-|t|2H)+,0<H≤12,
because S1/2(t) is known as the Slepian process and SH(t)-SH(0),0<t≤1, is equal in distribution to the fractional Brownian motion on the interval (0,1). By the Polya criterion, the fractional Slepian process exists because BSH(t) is a nonincreasing and a convex function on the semiaxis. The fact of the correlation function being nonnegative guarantees the existence of θ~SH in (1.4). SH(t) can be useful as a reference process in estimation of the survival exponents. Therefore it is important to have accurate estimates of the exponent for SH(t). The case of small H is the most interesting because it describes a transition of SH(t) to white noise. Our estimates of θ~SH are based on two lemmas, where we use the following notation:
(2.15)θ~(f,Δ)=-|Δ|-1logP(x(t)≤f(t),t∈Δ).Lemma 2.5 (see [12]).
Letx(t) be a centered Gaussian stationary process with a finite nonnegative correlation function, that is, Bx(t)≥0 and Bx(t)=0 for |t|≥T0. Then the limit
(2.16)θ~(a)=limT→∞θ~(a,(0,T)),
exists for every a∈R1. Moreover,
(2.17)(1+1k)-1θ~(a,kΔ0)≤θ~(a)≤θ~(a,kΔ0),Δ0=(0,T0).Remark 2.6.
Lemma1.1 was derived by Li and Shao [12] for the Slepian process, S1/2(t), but the proof remains valid for the general case. There is an explicit but very complicated formula for θ~SH(0,Δ) with H=1/2 [18]. In case of Δ=(0,2), this result reduces to
(2.18)P(S1/2(t)≤0,t∈(0,2))=16-2+38π
and gives 1.336<θ~S1/2<2.004.Lemma 2.7 (see [8]).
Letx(t) be a centered Gaussian process in an interval Δ with a correlation function B(t,s) and (Hx(Δ),∥·∥Δ) be the Hilbert space with the reproducing kernel B(t,s) on Δ×Δ. If 0<θ~(a,Δ)<∞, then
(2.19)|θ~(a+f,Δ)-θ~(a,Δ)|≤∥f∥Δ2|Δ|.Remark 2.8.
Lemma2.7 is a version of Proposition  1.6 from the paper by Aurzada and Dereich [8]; relation (2.19) successfully supplements the original Lemma 1.1.Proposition 2.9.
The persistence exponent of processSH(t) has the following estimates:
(2.20)-(1-H)H-1ln(2H)≤θ~SH≤49H-2,
where the left inequality holds for 0<H≤e-2/2.Corollary 2.10.
IfwH-(t)=(wH(t)-wH(-t))/2 is the odd component of the fractional Brownian motion, then
(2.21)θ~wH-≤(7/H)22,0<H<0.5.Proof.
The dual stationary processw~H- has the correlation function
(2.22)B~wH-(t)=(cosht2)2H-(sinht2)2H,
which is positive. Hence the exponent θ~wH- exists. The inequality
(2.23)B~wH-(2t)=(cosht)2H(1-(tanht)2H)≥(cosht)2H(1-|t|2H)+≥B~SH(t),
and Proposition 2.9 immediately imply the corollary.Remark 2.11.
The following estimates ofθ~wH- are due to Krug et al. [19]:
(2.24)θ~wH-≥min((1-H)2H,(1-H)21/(2H)-1),0<H<0.5,θ~wH-≤(1-H)2H,0.1549<H<0.5.
For small H these estimates are one-sided only.Remark 2.12.
A considerable difference in the behavior ofθ~wH- and θ~wH=1-H for small H is expected. Heuristically this can be explained as follows. As H→0, the discrete processes w~H-(kΔ) and w~H(kΔ) have different weak limits: {ξk} and {ξk-η/2}, respectively, where {ξk} and η are independent standard Gaussian variables. The probability (1.4) for the limiting processes is quite different:(2.25)P(ξk<0,k=1÷N)=2-N,P(ξk-η≤0,k=1÷N)=(N+1)-1.
Unfortunately, this argument fails to predict the behavior of θ~SH for small H, because the step Δ cannot be arbitrary and is a function of H.


## 2.4. Khanin’s Problem

The survival exponent for fractional Brownian motion in the intervalsΔT=(-T,T) is independent of the parameter H: θwH=1. This interesting fact follows from both self-similarity of wH and the stationarity of its increments [1].In the caseH<0.5, the variables wH(t) and wH(-t) are positive correlated. Therefore, a possible power-law asymptotics
(2.26)P(wH(t)<1,-wH(-t)<1,t∈(0,T))=T-θ+o(1),
where we change sign before wH(t) for negative t only, may have a radically different exponent compared with θwH=1. The question of finding bounds on the exponent θχH for the process
(2.27)χH(t)=sign(t)wH(t),ΔT=(-T,T),
was asked by K. Khanin. The next proposition contains a partial answer to this question.Proposition 2.13.
(
1
)
In the case 0.5≤H<1, the exponent θχH for ΔT=(-T,T) exists and admits of the following estimates:
(2.28)1<θχH(1-H)-1≤2,0.5≤H<1,in addition,θχ1/2=1.(2)
Letθ_χH be the lower exponent in (2.26), then
(2.29)θ_χH(1-H)-1≥(H-1-1)∧21/2H-1,0<H<0.25,θ_χH(1-H)-1≥20.25<H≤0.5.Remark 2.14.
To clarify whyθχH/θwH is unbounded for small H in the case ΔT=(-T,T), we consider again the limiting sequence for wH(kΔ) as H→0. This is {(ξk-ξ0)/2}, where the {ξk} are independent standard Gaussian variables. The probability (1.1) for the limit sequence is
(2.30)P{ξk<ξ0+2,|k|≤N}=(2N+1)-1l(N),
where l(N) is a slowly varying function, whereas for the limit sequence of χH(kΔ) we have
(2.31)P{ξ-k-2<ξ0<ξk+2,0<k≤N}≈πeN-1/2Φ(2)2N,
where Φ(x) is the Gaussian distribution function. As in Remark 2.12, we have nontrivial exponential asymptotics where the threshold for {ξk} is constant or bounded. Indeed, if the event in (2.31) is true, then
(2.32)|ξ0|<2+max(|∑1Nξ-k|,|∑1Nξk|)N=2+Op(1)N.


## 2.5. An Explicit Value ofθx

We have two explicit but isolated results for the fractional Brownian motion:θwH=(1-H) for ΔT=(0,T) and θwH=1 for ΔT=(-T,T). These results can be combined as follows.Proposition 2.15.
IfΔT=(-Tα,T), 0≤α≤1, then θwH=αH+(1-H).Remark 2.16.
The result is based on the following properties of the positiontΔ* of the maximum of wH(t) in Δ=[0,1]: tΔ* has a continuous probability density fΔ*(t) in (0, 1) and fΔ*(t)≈O(t-H) as t→0. In the case of multidimensional time, the behavior of fΔ*(t),Δ=(0,1)d near t=0 is a key to the survival exponent of wH(t) for ΔT=(-Tα,T)d,0<α<1 and H<1. By (1.2), θwH=d in the case α=1, and θwH=αd in the degenerate case: H=1.


## 3. Proofs

Proof of Proposition2.9
Lower Bound. Let w~H(t) be a dual fractional Brownian motion with the parameter H, that is, a Gaussian stationary process with correlation function B~wH(t)=cosh(Ht)-0.5(2sinh(t/2))2H. We prove in the appendix that for 0<H≤e-2/2,
(3.1)B~wH(pt)≥BSH(t),p=-H-1ln(2H).
Applying Slepian’s lemma, one has θ~SH≥p(1-H) because θ~wH=(1-H).
Upper Bound. The random variable η=∫01SH(t)dt corresponds to an element fη(t) of the Hilbert space, HS(Δ),Δ=(0,1), with the reproducing kernel B(t,s)=1-|t-s|2H. By definition of HS(Δ), we have
(3.2)fη(t)=ESH(t)η=1-t1+2H+(1-t)1+2H1+2H,∥fη∥Δ2=Eη2=H(3+2H)(1+H)-1(1+2H)-1.
It is easy to see thatfη(0)≤fη(t)≤fη(1/2). Therefore,
(3.3)H<fη(t)<2Hln(2e),4H3<∥fη∥Δ2<3H.
Let mH be the median of the random variable M=max{SH(t),t∈Δ}, where Δ=(0,1). Then
(3.4)0.5=P(SH(t)<mH,t∈Δ)<P(SH(t)<mHH-1fη(t),t∈Δ),
because H-1fη(t)>1. Setting x(t)=SH(t) in Lemma 2.5 and using notation (2.15), one has
(3.5)θ~(mHH-1fη(t),Δ)<ln2,θ~(0,Δ)<ln2+mHH-1∥fη∥Δ2.
Using Lemma1.1 and the inequality ∥fη∥Δ<3H, we have
(3.6)θ~SH<θ~(0,Δ)<(ln2+mH1.5H)2.
It is well known (see, e.g., [20]) that mH<42D(Δ,σ/2), where σ2=maxΔESH(t) and D is the Dudley entropy integral related to the semimetrics on Δ: ρ2(t,s)=E(SH(t)-SH(s))2.
In our caseρ(t,s)=2|t-s|H,σ=1 and therefore
(3.7)mH<cHH,
where
(3.8)cH=4(1-H)ln2+23-HπΦ(-1-Hln4)<5.36,H<12,
and Φ(x) is the standard Gaussian distribution. Hence,
(3.9)θ~SH<(ln2+5.361.5H)2<(7H)2.Proof of Proposition2.13
Part(1). In the case of H≥0.5, the process χH(t)=sign(t)wH(t) has nonnegative correlations on R1. In the standard manner, this implies the existence of θχH for ΔT=(-T,T). More precisely, starting from a self-similar 2D process x(t)=(wH(t),-wH(-t)) on R+1, we consider the dual 2D stationary process x~(t)=x(et)exp(-Ht) whose correlation matrix has positive elements. By [12], we conclude that the exponent θ~χH for x~(t) exists.
The equality
θ
~
χ
H
=
θ
χ
H
for
Δ
T
=
(
-
T
,
T
). We will use Lemma 1.1. By the relation χH(t)=sign(t)wH(t), the map φ(t)↦sign(t)φ(t) is an isometry between the Hilbert spaces HχH(ΔT)  and HwH(ΔT) associated with χH(t) and wH(t) on ΔT=(-T,T),respectively. To prove the equality of the dual exponents, it is enough to find φ(t)∈(HwH(R1),∥·∥R) such that sgn(t)φ(t)≥1 for |t|≥1. We can use
(3.10)φ(t)=sgn(t)min(|t|,1)=∫(eitλ-1)sinλiπλ2dλ,
because
(3.11)∥φ∥R2=kH∫(sinλ)2(πλ2)2|λ|1+2Hdλ<∞,
(see [14]).
Estimation of
θ
χ
H
,
H
>
1
/
2. Since EχH(t)χH(s)≥0 for any t, s, we have, by Slepian’s lemma,
(3.12)pT:=P(wH(t)<1,-wH(-t)<1,t∈(0,T))≥[P(wH(t)<1,t∈(0,T))]2.
Using (1.2), one has θχH≤2(1-H).
Obviously,pT≤P(wH(t)<1,t∈(0,T)). Therefore, θχH≥(1-H) for any H.
Part
(
2
)
.Let 0<H≤1/2, then EwH(t)(-wH(-s))≤0 for t,s>0. Hence,
(3.13)pT≤[P(wH(t)<1,t∈(0,T))]2,θχH≥2(1-H).
Finally,
(3.14)pT≤P(wH(t)-wH(-t)<2,t∈(0,T))=P(wH-(t)<1,t∈(0,T)).
But then, θ_χH≥θwH- for all H. If θwH-=θ~wH-, then we get a lower bound of θ~χH for 0<H≤1/4.
The equality
θ
w
H
-
=
θ
~
w
H
-.Let HwH-(Δ) and HwH(Δ) be the reproducing Kernel Hilbert spaces associated with wH_(t) and wH(t), respectively. By the definition of wH_(t), the map (φ(t),t>0)↦(sign(t)φ(|t|),|t|<∞) is an isometric embedding of HwH-(R+1) in HwH(R1). To prove that the exponents are equal, it is enough to find φ(t),t≥0 such that sign(t)φ(|t|)∈(HwH(R1),∥·∥R), φ(t)≥1 for t≥1, and ∥φ∥R<∞. As we showed above, this can be φ(t)=min(t,1),t>0.Proof of Proposition2.15
Consider the fractional Brownian motion inΔT=(-Tα,T),0≤α≤1. By Lemma 1.1, we can focus on the exponent related to the position of the maximum of wH(t) in ΔT,tΔT*.
Distribution of
t
Δ
*. We remind the main properties of the distribution function, F*(x), of tΔ* related to the normalized interval Δ=(0,1) (see [1, 14]):(i)
F*(x) has a continuous density fΔ*(x)>0,0<x<1 such that (1-x)fΔ*(x) decreases and xfΔ*(x) increases on Δ;(ii)
F*(x) have the following estimates:
(3.15)x1-Hl-1(x)≤F*(x)≤x1-Hl(x),wherel(x)=exp(c-lnx),c>0.
Due to monotonicity of(1-x)fΔ*(x) and xfΔ*(x), one has
(3.16)(1-x)fΔ*(x)≤x-1∫0x(1-u)fΔ*(u)du≤x-1F*(x),(3.17)xfΔ*(x)≥x-1∫xqxufΔ*(u)du≥q(F*(x)-F*(xq)),0<q<1.
By (3.15), (3.16),
(3.18)fΔ*(x)≤x-Hl(x)(1-x)-1.
Using (3.15), (3.17), one has
(3.19)fΔ*(x)≥qx-Hl-1(x)(1-l(x)l(xq)q1-H).
If we set q1-H=l-2(x)/2, then
(3.20)fΔ*(x)≥qx-Hl-1(x)2=cHx-Hl-νH(x),
where νH=(3-H)/(1-H), cH=2-(2-H)/(1-H).
Distribution of
t
Δ
T
*. Let T1=T-+T, where T-=Tα, then the processes wH(T1τ-T-)-wH(-T-)and wH(τ)T1H on Δ=(0,1) are equal in distribution. Hence, tΔT* and T1tΔ*-T- have the same distribution as well. Therefore,
(3.21)pT:=P(|tΔT*|≤0.5)=P(|tΔ*-T-T1|≤0.5T1)=T1-1fΔ*(T-+εT1),
where |ε|≤0.5. We have used here the existence and continuity of fΔ*(x).
Exponent
θ
˘
w
H
.Set α=1. Then (3.21) implies limT→∞TpT=0.5fΔ*(0.5).
Letα<1, then (T-+ε)/T1=o(1) as T→∞, and (3.20), (3.21) give a lower bound on pT:
(3.22)T1pT≥cH(aT+)-Hl-νH(aT+).
Here and below aT±=(T-±0.5)/T1.
Using (3.18), (3.21), we get an upper bound on pT:
(3.23)T1pT=fΔ*(Tα+εT1)≤(aT-)-Hl(aT-)T1T+1≤2(aT-)-Hl(aT-).
By substituting T-=Tα, we have
(3.24)lnaT±=-(1-α)lnT+O(T-β),β=α∧(1-α),lnl(aT±)=O(lnT).
Hence,
(3.25)-lnpT=(1-(1-α)H)lnT+O(lnT),
that is, θ˘wH=αH+(1-H).
The equality
θ
˘
w
H
=
θ
w
H. Consider the Hilbert space (HwH(R1),∥·∥R) related to FBM and a function
(3.26)φ(t)=min(|t|,1)=∫(eitλ-1)(sinλ/22πλ/2)2dλ.
The standard spectral representation of the kernel EwH(t)wH(s) and the representation (3.26) yield
(3.27)∥φ∥R2=kH∫(sinλ/22πλ/2)4|λ|1+2Hdλ<∞,
where kH=∫|eiλ-1|2|λ|-1-2H. Setting φT:={φ(t),t∈ΔT}, the desired statement follows from Lemma 1.1 because φT∈(HwH(ΔT),∥·∥T) and ∥φT∥T≤∥φ∥R.



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*Source: 137271-2012-11-29.xml*