Sojourn time dimensions of fractional Brownian motion
SSOJOURN TIME DIMENSIONS OFFRACTIONAL BROWNIAN MOTION
IVAN NOURDIN, GIOVANNI PECCATI, AND ST´EPHANE SEURET
Abstract.
We describe the size of the sets of sojourn times E γ = { t ≥ | B t | ≤ t γ } associated with a fractional Brownian motion B in terms of various large scale dimensions. Keywords:
Sojourn time; logarithmic density; pixel density; macro-scopic Hausdorff dimension; fractional Brownian motion.
AMS 2010 Classification: Introduction
Describing the properties of the sample paths of stochastic processesis one of the leading threads of modern stochastic analysis: such a lineof research started with the investigation of the almost sure continuityproperties of the paths of a real-valued Brownian motion ( B t ) t ≥ , suchas H¨older continuity, followed by the important notions of fast andslow points introduced by Taylor. See e.g. the three classical references[10, 16, 20] for formal statements, as well as for an historical overviewof this fundamental domain.A naturally connected question consists in describing the geometricproperties of the graph of { ( t, B t ) : t ≥ } , in terms of box , packing and Hausdorff dimensions – see Section 2.1 for precise definitons. Inthis respect, the case of the Brownian motion is [23, 24, 16] now verywell understood, and many researchers have tried, often succesfully, toobtain similar results for other widely used classes of processes: frac-tional Brownian motions and more general Gaussian processes, L´evyprocesses, solutions of SDE or SPDE’s (see [17, 18, 2, 27, 13, 22] for in-stance, and the numerous references therein). Despite these remarkableefforts, many important questions in this area are almost completelyopen for future research.
Date : September 5, 2018.S. Seuret thanks the RMATH at University of Luxembourg for its support duringhis stay in 2017-2018. a r X i v : . [ m a t h . P R ] S e p IVAN NOURDIN, GIOVANNI PECCATI, AND ST´EPHANE SEURET
Figure 1.
The red subset of the real line is a simulationof the set E γ , with γ = 0 . and H = 0 . sojourn times . The objective is to describe the (asymptotic) propor-tion of time spent by a stochastic process in a given region. Sojourntimes have been studied by many authors (see for instance [9, 19, 25, 8]and the references therein) and play a key role in understanding var-ious features of the paths of stochastic processes, especially those ofBrownian motion.In this paper, we focus on the sojourn times associated with thepaths of a fractional Brownian motion (FBM) inside the domain { ( t, u ) : t ≥ | u | ≤ t γ } , where γ ≥
0. It is known that, with probabilityone, after some large time t , an FBM B := ( B t ) t ≥ does not intersectthe domain { ( t, u ) : t ≥ | u | ≥ t H + ε } , for every ε >
0. For thisreason, in what follows we restrict the study to the case γ ∈ [0 , H ], andinvestigate the sets(1) E γ := { t ≥ | B t | ≤ t γ } in terms of various large scale dimensions: the Lebesgue density , the log-arithmic density , and the macroscopic box and
Hausdorff dimensions .A simulation of the set E γ appears in Fig. 1. OJOURN TIMES FOR FRACTIONAL BROWNIAN MOTION 3
The last two notions evoked above have been introduced in the late1980s by Barlow and Taylor (see [3, 4]), in order to formally define thefractal dimension of a discrete set. One of the main motivations for thetheory developed in [3] was e.g. to describe the asymptotic propertiesof the trajectory of a random walk on Z , whereas the focus in [4] wasthe computation of the macroscopic Hausdorff dimension of an α -stablerandom walk. Proper definitions are given in the next section. Thesedimensions have proven to be relevant in other situations, in particularwhen describing the high peaks of (random) solutions of the stochasticheat equation, see the seminal works of Khoshnevisan, Kim and Xiao[12, 14].The present paper can be seen as a follow-up and a non-trivial ex-tension of [21], where analogous results were obtained in the case of B being a standard Brownian motion. One of the principal motiva-tions of our analysis is indeed to understand how much the findingsof [21] rely on the specific features of Brownian motion, such as the(strong and weak) Markov properties, the associated reflection princi-ple, as well as the fine properties of local times. While all these featuresare heavily exploited in [21], the novel approach developed in our pa-per shows that the dimensional analysis of sojourn times initiated in[21] can be substantially extended to the non-Markovian setting of afractional Brownian motion with arbitrary Hurst index. We believethat our techniques might be suitably adapted in order to study so-journ times associated with even larger classes of Gaussian processesor Gaussian fields.From now on, every random object considered in the paper is definedon a common probability space (Ω , A , P ), with E denoting expectationwith respect to P .2. Assumptions and main results
Densities and dimensions.
In what follows, the symbol ‘Leb‘stands for the one-dimensional Lebesgue measure. For any set A , wedenote by | A | its cardinality whereas, for any subset E ⊂ R + ,pix( E ) = { n ∈ N : dist( n, E ) ≤ } is the set of integers that are at distance less than one from E . It isclear that Leb( E ) ≤ | pix( E ) | , while the converse inequality does nothold in general.We will now describe the main notions and concepts that are usedin this paper, in order to describe the size of the set of sojourn times E γ = { t ≥ | B t | ≤ t γ } . IVAN NOURDIN, GIOVANNI PECCATI, AND ST´EPHANE SEURET
The simplest way of assessing the size of E γ simply consists in esti-mating how fast the Lebesgue measure of E γ ∩ [0 , t ] grows with t . Fora general set E ⊂ R + , this yields the following definition. Definition 1.
Let E ⊂ R + . The logarithmic density of E is definedas Den log E = lim sup n → + ∞ log Leb( E ∩ [1 , n ]) n . This notion will be compared with a similar quantity, obtained byreplacing the Lebesgue measure of a given subset of E by the cardinalityof its pixel set. Definition 2.
Let E ⊂ R + . The pixel density of E is defined by Den pix E = lim sup n → + ∞ log | pix( E ∩ [1 , n ]) | n The last notion we will deal with is the macroscopic Hausdorff di-mension , introduced by Barlow and Taylor (as discussed above), inorder to quantify a sort of “fractal” behavior of self-similar structuressitting on infinite lattices.Following the notations of [12, 14], we consider the annuli S = [0 , S n = [2 n − , n ), for n ≥
1. For any ρ ≥
0, any set E ⊂ R + andany n ∈ N ∗ , we define ν nρ ( E ) = inf (cid:40) m (cid:88) i =1 (cid:18) Leb( I i )2 n (cid:19) ρ : m ≥ , I i ⊂ S n , E ∩ S n ⊂ m (cid:91) i =1 I i (cid:41) , (2)where I i are non-trivial intervals with integer boundaries (hence theirlength is always greater or equal than 1). The infimum is thus takenover a finite number of finite families of non-trivial intervals. Definition 3.
Let E ⊂ R + . The macroscopic Hausdorff dimension of E is defined as (3) Dim H E = inf (cid:40) ρ ≥ (cid:88) n ≥ ν nρ ( E ) < + ∞ (cid:41) . Observe that Dim H E ∈ [0 ,
1] for any E ⊂ R + : indeed, choosing ascovering of E ∩ S n the intervals of length 1 partitioning S n , we get ν n ε ( E ) ≤ − nε for any ε >
0, so that (cid:80) nε ν n ε ( E ) < + ∞ .The macroscopic Hausdorff dimension Dim H E of E ⊂ R + does notdepend on its bounded subsets, since the series in (3) converges if and OJOURN TIMES FOR FRACTIONAL BROWNIAN MOTION 5 only if its tail series converges. In particular, every bounded set E hasa macroscopic Hausdorff dimension equal to zero - the converse is nottrue, for instance Dim H (cid:83) n ≥ { n } = 0.Observe also that the local structure of E does not really influencethe value of Dim H ( E ), since the ”natural” scale at which E is observedis 1.The value of Dim H E describes the asymptotic distribution of E ⊂ R + on R + . The difference between Dim H E and the previously intro-duced dimensions is that while Den pix E (or Den log E ) only counts thenumber of points of E ∩ S n (or, equivalently, measures E ∩ [1 , n ]),the quantity Dim H E takes into account the geometry of the set E , inparticular by considering the most efficient covering of E ∩ S n . Forinstance, as an intuition, the value of ν ρn ( E ) is large when all the pointsof E ∩ S n are more or less uniformly distributed in S n , while it is muchsmaller when these points are all located in the same region (in thatcase, one large interval is the best possible covering).Standard inequalities exploited in our paper are ( see [3, 12])(4) Dim H E ≤ Den pix E and Den log E ≤ Den pix E. These inequalities are strict in general, in particular the first one willbe strict for the sets we focus on in this paper.2.2.
Fractional Brownian motion.
Throughout the paper, B =( B t ) t ≥ denotes a one-dimensional fractional Brownian motion (FBM)of index H ∈ (0 , B is a continuous Gaussian pro-cess, centered, self-similar of index H , and with stationary increments.All these properties (in particular, the fact that one can always select acontinuous modification of B ) are simple consequences of the followingexpression for its covariance function R : R ( u, v ) = E [ B u B v ] = 12 (cid:0) u H + v H − | v − u | H (cid:1) . One can easily check that(5) I := (cid:90) (cid:90) [0 , du dv (cid:112) R ( u, u ) R ( v, v ) − R ( u, v ) < + ∞ . By virtue of this fact, the local time ( L xt ) x ∈ R ,t ≥ associated with B iswell defined in L (Ω) by the following integral relation:(6) L xt = 12 π (cid:90) R dy e − iyx (cid:90) ts du e iyB u , IVAN NOURDIN, GIOVANNI PECCATI, AND ST´EPHANE SEURET see e.g. [6]. For each t , the local time x (cid:55)→ L xt is the density of theoccupation measure µ t ( A ) = Leb { s ∈ [0 , t ] : B s ∈ A } associated with B . Otherwise stated, one has that L t = dµ t d Leb .A last property that we will need in order to conclude our proofs,and that is an immediate consequence of the Volterra representation of B , is that the natural filtration associated with FBM is Brownian. Bythis, we mean that there exists a standard Brownian motion ( W u ) u ≥ defined on the same probability space than B such that its filtrationsatisfies(7) σ { B u : u ≤ t } ⊂ σ { W u : u ≤ t } . for all t > Our results.
Let the notation of the previous sections prevail(in particular B denotes a FBM of index H ∈ (0 , E γ , as defined in (1). Theorem 1.
Fix γ ∈ [0 , H ) . Then (8) Den pix E γ = Den log E γ = γ + 1 − H a.s. Our second theorem deals with the macroscopic Hausdorff dimensionof all sets E γ . Theorem 2.
Fix γ ∈ [0 , H ) . Then (9) Dim H E γ = 1 − H a.s. The fact that the macroscopic box and Hausdorff dimension differasserts that the trajectory enjoys some specific geometric properties.This can be interpreted by the fact that the set E γ is not uniformlydistributed (if it were, then both dimensions would coincide), whichrelies on the intuition that the trajectory of an FBM does not fluctuatetoo rapidly from one region to the other.Actually, the lower bound for the dimension Dim H E γ ≥ − H inTheorem 2 will follow from the next statement, which evaluates thedimension of the level sets(10) L x := { t : B t = x } and which is of independent interest. Theorem 3.
Fix x ∈ R . Then (11) Dim H L x = 1 − H a.s. OJOURN TIMES FOR FRACTIONAL BROWNIAN MOTION 7
The connection between Theorem 2 and Theorem 3 can be heuristi-cally understood by observing that, when t is large and for a fixed x ,the relation t ∈ L x implies that t ∈ E γ , and therefore Dim H E γ ≥ Dim H L x , owing to the fact that, as explained above, the quantityDim H A does not depend on the bounded subsets of a given A ⊂ R + .3. Proof of Theorem 1 : values of
Den pix E γ and Den log E γ In what follows,
C >
Upper bounds.
Recalling the second part of (4), it is enough tofind an upper bound for Den pix E γ , which will also be an upper boundfor Den log E γ .Fix γ ∈ (0 , H ), and consider Den pix E γ . First, we observe that E ( | pix( E γ ) ∩ [1 , n ] | ) = n (cid:88) m =1 P ( ∃ s ∈ [ m − , m + 1] , | B s | ≤ s γ )= n (cid:88) m =1 P ( ∃ s ∈ [1 − m , m ] , | B s | ≤ s γ m γ − H ) ≤ n (cid:88) m =1 ( A − /m + A +1 /m )where A − ε := P ( ∃ s ∈ [1 − ε, , | B s | ≤ ε H − γ ) A + ε := P ( ∃ s ∈ [1 , ε ] , | B s | ≤ ε H − γ ) . Lemma 4.
For every ε small enough , (12) max( A − ε , A + ε ) ≤ ε H − γ . Proof.
Let us consider A − ε first. We have A − ε ≤ P ( | B | ≤ ε H − γ )+ P ( ∃ s ∈ [1 − ε, , | B s − B | ≥ ε H − γ ) . The term P ( | B | ≤ ε H − γ ) is easily bounded by Cε H − γ , so let usconcentrate on the term P ( ∃ s ∈ [1 − ε, , | B s − B | ≥ ε H − γ ). Set X s = B − B − s , s ∈ [0 , X is also a FBM. We have P ( ∃ s ∈ [1 − ε, , | B s − B | ≥ ε H − γ )= P ( ∃ s ∈ [0 , , | X εs | ≥ ε H − γ ) = P ( ∃ s ∈ [0 , , | X s | ≥ ε − γ )= P ( sup s ∈ [0 , | X s | ≥ ε − γ ) ≤ P ( sup s ∈ [0 , X s ≥ ε − γ ) IVAN NOURDIN, GIOVANNI PECCATI, AND ST´EPHANE SEURET where last inequality makes use of the fact that X law = − X . It is well-known that, by virtue of the Borell and Tsirelson-Ibragimov-Sudakovinequalties (see e.g [1, Section 2.1]), setting α = E (cid:2) sup [0 , B (cid:3) andbecause E [ B s ] = s H ≤ s ∈ [0 , P (sup [0 , X ≥ u ) ≤ e − ( u − α )22 , u ≥ . (That α is finite is part of the result.) We deduce that P ( ∃ s ∈ [1 − ε, , | B s − B | ≥ ε H − γ ) ≤ e − ( ε − γ − α )22 = O ( ε δ ) , for every δ > ε becomes small enough. Hence the result. Ananalogous argument leads to the same estimate for the set A + ε . (cid:3) Going back to A − /m and A +1 /m , we obtain from Lemma 4 thatmax( A − /m , A +1 /m ) = O ( m γ − H ) . We consequently conclude that E ( | pix( E γ ) ∩ [1 , n ] | ) ≤ n (cid:88) m =1 ( A − /m + A +1 /m ) = O (2 n ( γ +1 − H ) ) . Choosing ρ > γ + 1 − H , we have (cid:88) n ≥ P ( | pix( E γ ) ∩ [1 , n ] | > nρ ) ≤ C (cid:88) n ≥ n (1+ γ − H ) nρ < + ∞ . Using the Borel-Cantelli lemma we infer that, with probability one, | pix( E γ ) ∩ [1 , n ] | ≤ nρ for every large enough integer n . Hence Den pix E γ ≤ ρ . Letting ρ ↓ γ + 1 − H leads to Den pix E γ ≤ γ + 1 − H . (cid:3) Remark 1.
We could have proved directly the upper bound for Den log E γ as follows. Introduce(14) S γ ( t ) = Leb { ≤ s ≤ t : | B s | ≤ s γ } . Its expectation can be estimated : E ( S γ ( t )) = (cid:90) t P ( | B s | ≤ s γ ) ds = (cid:90) t P ( | B | ≤ s γ − H ) ds ∼ Ct γ +1 − H , (15)where the Fubini theorem, the self-similarity of B and then the fact that B ∼ N (0 ,
1) have been successively used. The same argument (based
OJOURN TIMES FOR FRACTIONAL BROWNIAN MOTION 9 on Borel-Cantelli) as the one used above to conclude that Den pix E γ ≤ γ + 1 − H , allows one to deduce the desired result.3.2. Lower bounds.
To obtain the announced lower bounds, we firstevaluate the second moment of S γ ( t ) (defined in (14)). We have, withΣ u,v denoting the covariance matrix of ( B u , B v ), E ( S γ ( t ) )= (cid:90) (cid:90) [0 ,t ] P ( | B u | ≤ u γ , | B v | ≤ v γ ) dudv = t (cid:90) (cid:90) [0 , P ( | B u | ≤ u γ t γ − H , | B v | ≤ v γ t γ − H ) dudv = t π (cid:90) (cid:90) [0 , dudv (cid:112) det Σ u,v (cid:90) (cid:90) R e − ( x,y ) T Σ − u,v ( x,y ) (cid:26) | x | ≤ u γ t γ − H | y | ≤ v γ t γ − H (cid:27) dxdy. Upper bounding e − { ... } , u and v by 1 and using that (5) is satisfied,we deduce that(16) E ( S γ ( t ) ) ≤ C t γ +2 − H . Applying the Paley-Zygmund inequality together with the estimate(15), we deduce from (16) that, for any fixed 0 < c <
1, there exists c (cid:48) > P ( S γ (2 n ) ≥ c n ( γ +1 − H ) ) ≥ (1 − c ) E ( S γ (2 n )) E ( S γ (2 n ) ) ≥ c (cid:48) . The Borel-Cantelli lemma ensures that, for infinitely many integers n , S γ (2 n ) ≥ c n ( γ +1 − H ) . This fact implies that Den log E γ ≥ γ + 1 − H .Finally, using the right inequality in (4), we directly obtain Den pix E γ ≥ γ + 1 − H .4. Proof of Theorem 2: value of Dim H E γ Upper bound for
Dim H E γ . In what follows, c > ≤ γ < H , as well as η > H E γ ≤ − H + η . Letting η tend to zerowill then give the result.Fix ρ > − H + η . Consider for every integer n ≥ i ∈{ , ..., (cid:98) n − / n γH (cid:99)} the times t n,i = 2 n − + i n γH . The collection t n,i generates the intervals I n,i = [ t n,i , t n,i +1 ), togetherwith the associated event E n,i = {∃ t ∈ I n,i : | B t | ≤ t γ } , Set ε n,i = 2 n γH /t n,i , so that I n,i = [ t n,i , t n,i (1 + ε n,i )), and observe thatthe ratio between any two of the quantities 2 n ( γH − , ε n,i and t γH − n,i arebounded uniformly with respect to n and i . By self-similarity, we havethat, when n becomes large, P ( E n,i ) = P ( ∃ τ ∈ [1 , ε n,i ] : | B τ · t n,i | ≤ ( τ · t n,i ) γ )= P ( ∃ τ ∈ [1 , ε n,i ] : | B t | ≤ t γ − Hn,i τ γ ) ≤ P ( ∃ τ ∈ [1 , ε n,i ] : | B t | ≤ t γ − Hn,i ) ≤ P ( ∃ τ ∈ [1 , ε n,i ] , | B t | ≤ c ε Hn,i ) ≤ P ( ∃ t ∈ [1 , ε n,i ] , | B t | ≤ ε H − ηn,i ) . The last estimate holds because η is a small positive real number and ε n,i tends to zero. By Lemma 4, we deduce that P ( E n,i ) ≤ c ε H − ηn,i andthen P ( E n,i ) ≤ c n ( γ − H ) H − ηH . Now observe that E n,i is realized if and only if E γ ∩ I n,i (cid:54) = ∅ . So, usingthe intervals I n,i as a covering of E γ ∩ I n,i (cid:54) = ∅ , we obtain from (2) that E [ ν nρ ( E γ )] ≤ E (cid:98) n − − n γH (cid:99) (cid:88) i =0 (cid:18) Leb( I n,i )2 n (cid:19) ρ E n,i ≤ ρn ( γH − (cid:98) n − − nγ/H (cid:99) (cid:88) i =0 P ( E n,i ) ≤ c n H − γH (1 − H + η − ρ ) . Thus, the Fubini Theorem entails E [ (cid:80) ∞ n =1 ν nρ ( E γ )] < + ∞ as soon as ρ > − H + η . This implies that for such ρ ’s, the sum (cid:80) ∞ n =1 ν nρ ( E γ ) isfinite almost surely. In particular, Dim H E γ ≤ ρ for every ρ > − H + η .Since such a relation holds for an arbitrary (small) ρ >
0, we deducethe desired conclusion.4.2.
Lower bound Dim H E γ ≥ − H . This lower bound follows fromthe lower bound in Theorem 3, as proved in Section 5.3.Indeed, assume that Dim H L ≥ − H , which is an almost sureconsequence of Theorem 3. Obviously L ⊂ E γ , hence Dim H E γ ≥ − H , which is the desired conclusion. OJOURN TIMES FOR FRACTIONAL BROWNIAN MOTION 11 Proof of Theorem 3
A slight modification of Dim H E . In this section, we use aslightly modified version of ν nρ defined as˜ ν nρ ( E ) = inf (cid:40) m (cid:88) i =1 (cid:18) Leb( I i )2 n (cid:19) ρ (cid:12)(cid:12)(cid:12)(cid:12) log Leb( I i )2 n (cid:12)(cid:12)(cid:12)(cid:12) − ρ : m ≥ , I i ⊂ S n , E ∩ S n ⊂ m (cid:91) i =1 I i (cid:41) . (17)The introduction of a logarithm factor makes some computations easierin Section 5.3. The quantities ν nρ lead to the same notion of dimension.Indeed, it is easily proved [12, 21] that one can replace ν by ˜ ν in (3),so that(18) Dim H E = inf (cid:40) ρ ≥ (cid:88) n ≥ ˜ ν nρ ( E ) < + ∞ (cid:41) . Upper bound for Dim H L x . The argument exploited in thepresent section is comparable to the one used in Section 4.2.Since every levet set L x defined by (10) is ultimately included in E γ for every γ >
0, and since all the dimensions we consider do not dependof any bounded subset of E γ , we easily obtain from (4) and Theorem1 that Dim H L x ≤ − H + γ , for any γ >
0. Letting γ ↓
0, one seesthat Dim H L x ≤ − H .5.3. Lower bound for Dim H L x . Let us now introduce the randomvariables(19) Y xn = L x nH n − L x nH n − n (1 − H ) and F xN := N (cid:88) n =1 Y xn . The random sequence ( F xN ) N ≥ is non-decreasing and we denote by F x ∞ its limit, i.e. F x ∞ = (cid:80) n ≥ Y xn .We remark from the self-similarity of B that Y xn d = Y x , see formula (6).Let us start with a lemma connecting the r.v. Y xn to the macroscopicHausdorff dimension. Lemma 5.
With probability one, there exists a constant
K > suchthat, for every x ∈ R and every n ≥ , ˜ ν n − H ( L x ) ≥ K − Y xn . Proof.
We start by recalling a key result of Xiao (see [26, Theorem 1.2]),which describes the scaling behavior of the local times of stationaryGaussian processes. For this, let us introduce the random variables X n := sup ≤ t ≤ n sup ≤ h ≤ n − sup x ∈ R L xt + h − L xt h − H ( n − log h ) H . Self-similarity of B implies X n = sup ≤ t ≤ sup ≤ h ≤ / sup x ∈ R L x nH n ( t + h ) − L x nH n t (2 n h ) − H ( − log h ) Hd = sup ≤ t ≤ sup ≤ h ≤ / sup x ∈ R L xt + h − L xt h − H ( − log h ) H By [26, Theorem 1.2], with probability one there exists a constant
K > n ≥ X n ≤ K. Now fix x ∈ R , and consider the associated level set L x defined by (10).Recall the definition (17) of ˜ ν n − H ( L x ). Choose a covering ( I i ) i =1 ,...,m that minimizes the value in (17), and set I i = [ x i , y i ]. We observe that˜ ν n − H ( L x ) = m (cid:88) i =1 (cid:18) Leb( I i )2 n (cid:19) − H (cid:12)(cid:12)(cid:12)(cid:12) log Leb( I i )2 n (cid:12)(cid:12)(cid:12)(cid:12) H = m (cid:88) i =1 (cid:18) | y i − x i | n (cid:19) − H (cid:12)(cid:12)(cid:12)(cid:12) log | y i − x i | n (cid:12)(cid:12)(cid:12)(cid:12) H ≥ K − m (cid:88) i =1 L x nH y i − L x nH x i n (1 − H ) = K − m (cid:88) i =1 L x nH ( I i )2 n (1 − H ) ≥ K − L x nH n − L x nH n − n (1 − H ) where (20) has been used to get the first inequality, and the last in-equality holds because the local time L x. increases only on the sets I i (whose union covers L x ∩ S n ). This proves the claim. (cid:3) Remark 2.
The introduction of ˜ ν nρ instead of ν nρ in (18) is key in thelast sequence of inequalities displayed in the previous proof, allowingus to use in a relevant way Xiao’s result (20).Now, using Lemma 5, and recalling (18), in order to conclude thatDim H L x ≥ − H and Theorem 3, it is enough to prove that theseries (cid:80) n ≥ Y xn diverges almost surely. This is the purpose of the nextproposition. OJOURN TIMES FOR FRACTIONAL BROWNIAN MOTION 13
Proposition 6.
For all x ∈ R , (21) P ( F x ∞ = + ∞ ) = 1 . The proof of Proposition 6 makes use of various arguments involv-ing local times, Brownian filtration and Kolmogorov 0-1 law. As apreliminary step, we start with the following lemma, showing that theprevious probability is strictly positive. Our key argument can be seenas a variation of the celebrated
Jeulin’s Lemma [15, p. 44], allowingone to deduce the convergence of random series (or integrals), by con-trolling deterministic series of probabilities.
Lemma 7.
For every x ∈ R , one has that (22) P ( F x ∞ = + ∞ ) > . Proof.
Recalling (6) we have, for every s ≤ t , L xt − L xs = 12 π (cid:90) R dy e − iyx (cid:90) ts du e iyB u . Using the self-similarity of B through E ( B u ) = u H E ( B ) = u H , wededuce: E ( L xt − L xs ) = 12 π (cid:90) R dy e − iyx (cid:90) ts du e − y u H = 12 π (cid:90) ts du (cid:90) R dy e − iyx e − y u H = 1 √ π (cid:90) ts e − x u H u − H du. We observe in particular that E ( L x − L x ) >
0, so(23) P ( Y x >
0) = P ( L x − L x > > . Now fix γ >
0, and consider the event A = { F x ∞ ≤ γ } . We have, byFubini, γ ≥ E ( A F x ∞ ) = (cid:88) n ≥ E ( A Y xn ) = (cid:88) n ≥ (cid:90) + ∞ P ( A ∩ { Y xn > u } ) du. Using P ( A ∩ B ) ≥ ( P ( A ) − P ( B c )) + , we deduce that γ ≥ (cid:88) n ≥ (cid:90) + ∞ ( P ( A ) − P Y xn ≤ u )) + du = (cid:88) n ≥ (cid:90) + ∞ ( P ( A ) − P ( Y x ≤ u )) + du. Since the summand does not depend on n , the only possibility is thatit is zero, that is, (cid:90) + ∞ ( P ( F x ∞ ≤ γ ) − P ( Y x ≤ u )) + du = 0 . This implies, for almost every u ≥ γ > P ( F x ∞ ≤ γ ) ≤ P ( Y x ≤ u ) . Letting γ → + ∞ together with u → + , and recalling (23), we concludethat P ( F x ∞ = + ∞ ) ≥ P ( Y x > > , which is exactly the desired relation (22). (cid:3) It remains us to prove that not only P ( F x ∞ = + ∞ ) is strictly positivefor every x , but in fact it equals 1. Such a conclusion will follow from thenext statement, corresponding to a time-inversion property of FBM. Itcan be checked immediately by computing the covariance function ofthe process (cid:101) B introduced below. Lemma 8.
The reversed time process (cid:101) B (24) t (cid:55)−→ (cid:101) B u := u H B /u is also a FBM. Let us denote by (cid:101) L xt , (cid:101) Y xn and (cid:101) F xN the quantities analogous to L xt , Y xn and F xN defined in (19), but associated with (cid:101) B (see (24)) instead of B .Obviously, ( L xt ) x ∈ R ,t ≥ and ( (cid:101) L xt ) x ∈ R ,t ≥ have the same law. So F x ∞ d = (cid:101) F x ∞ := + ∞ (cid:88) n =1 (cid:101) L nH x n − (cid:101) L nH x n − n (1 − H ) . For a fixed integer n ≥
1, we have (cid:101) L x nH n − (cid:101) L x nH n − = 12 π (cid:90) R dy e − iy nH x (cid:90) n n − du e iyu H B /u , implying in turn that (cid:101) L x nH n − (cid:101) L x nH n − is σ { B u : u ≤ − ( n − } -measurable.As a consequence, for every M ≥ σ (cid:110)(cid:101) L x nH n − (cid:101) L x nH n − : n ≥ M (cid:111) ⊂ σ { B u : u ≤ − ( M − } . The event { (cid:101) F x ∞ = + ∞} does not depend on the first term of the series,so is a tail event. Otherwise stated, { (cid:101) F x ∞ = + ∞} ∈ (cid:92) M ≥ σ (cid:110)(cid:101) L x nH n − (cid:101) L x nH n − : n ≥ M (cid:111) . OJOURN TIMES FOR FRACTIONAL BROWNIAN MOTION 15
Using now (7) (with (cid:101) B instead of B ), we deduce that { (cid:101) F x ∞ = + ∞} ∈ (cid:92) M ≥ σ { W u : u ≤ − M } , where W is a standard Brownian motion. By the Blumenthal’s 0-1law for W , we infer that P ( (cid:101) F x ∞ = + ∞ ) is either 0 or 1. RememberingLemma 7, we can conclude that this probability is one, which implies(21) as claimed. (cid:3) Acknowledgement . We thank Ciprian Tudor for suggesting thatit may be useful to rely on the representation (6) of the local time,and to Yimin Xiao for useful discussions around the (wide) literaturegenerated by his seminal work [26].
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In press. DOI: 10.1007/s10959-017-0784-y
Ivan Nourdin, Universit´e du Luxembourg, Unit´e de Recherche enMath´ematiques, Maison du Nombre, 6 avenue de la Fonte, L-4364 Esch-sur-Alzette, Grand Duch´e du Luxembourg
E-mail address : [email protected] Giovanni Peccati, Universit´e du Luxembourg, Unit´e de Rechercheen Math´ematiques, Maison du Nombre, 6 avenue de la Fonte, L-4364Esch-sur-Alzette, Grand Duch´e du Luxembourg
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