Sample Path Properties of Bifractional Brownian Motion

Transcription

1 Sample Path Properties of Bifractional Brownian Motion Ciprian A. Tudor SAMOS-MATISSE, Centre d Economie de La Sorbonne, Université de Panthéon-Sorbonne Paris, 9, rue de Tolbiac, Paris Cedex 3, France. tudor@univ-paris.fr URL: Yimin Xiao Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USA. xiao@stt.msu.edu URL: October 22, 27 Abstract Let B H,K = { B H,K t), t R + } be a bifractional Brownian motion in R d. By connecting it to a stationary Gaussian process through Lamperti s transform, we prove that B H,K is strongly locally nondeterministic. Applying this property and a stochastic integral representation of B H,K, we establish Chung s law of the iterated logarithm for B H,K, as well as sharp Hölder conditions and tail probability estimates for the local times of B H,K. We also consider the existence and regularity of the local times of multiparameter bifractional Brownian motion B H,K = { B H,K t), t R N + } in R d using Wiener-Itô chaos expansion. Running head: Sample Path Properties of Bifractional Brownian Motion 2 AMS Classification Numbers: Primary 6G5, 6G7. Key words: Bifractional Brownian motion, self-similar Gaussian processes, small ball probability, Chung s law of the iterated logarithm, local times, level set, Hausdorff dimension, chaos expansion, multiple Wiener-Itô stochastic integrals. Research partially supported by the NSF grant DMS changed gaussian to Gaussian

2 Introduction In recent years, there has been considerable interest in studying fractional Brownian motion due to its applications in various scientific areas including telecommunications, turbulence, image processing and finance. Many authors have also proposed using more general selfsimilar Gaussian processes and random fields as stochastic models; see e.g. Addie et al. 999), Anh et al. 999), Benassi et al. 2), Mannersalo and Norros 22), Bonami and Estrade 23), Cheridito 24), Benson et al. 26). Such applications have raised many interesting theoretical questions about self-similar Gaussian processes and fields in general. However, in contrast to the extensive studies on fractional Brownian motion, there has been little systematic investigation on other self-similar Gaussian processes. The main reasons for this, in our opinion, are the complexity of dependence structures and the non-availability of convenient stochastic integral representations for self-similar Gaussian processes that do not have stationary increments. The obective of this paper is to fill this gap by developing systematic ways to study sample path properties of self-similar Gaussian processes. Our main tools are the Lamperti transformation [which provides a powerful connection between self-similar processes and stationary processes; see Lamperti 962)] and the strong local nondeterminism of Gaussian processes [see Xiao 27)]. In particular, for any self-similar Gaussian process X = {Xt), t R}, the Lamperti transformation leads to a stochastic integral representation for X. We will show the usefulness of such a representation in studying sample path properties of X. To illustrate our methods, we only consider a rather special class of self-similar Gaussian processes, namely, the bifractional Brownian motions introduced by Houdré and Villa 23). Given constants H, ) and K, ], the bifractional Brownian motion bi-fbm, in short) in R is a centered Gaussian process B H,K = {B H,K t), t R + } with covariance function R H,K s, t) := Rs, t) = [ 2 K t 2H + s 2H ) K t s 2HK].) and B H,K ) =. Let B H,K,..., B H,K d B H,K = { B H,K } t), t R + with values in R d by B H,K t) = B H,K be independent copies of B H,K. We define the Gaussian process t),..., B H,K d ) t), t R +..2) By.) one can verify easily that B H,K is a self-similar process with index HK, that is, for every constant a >, {B H,K at), t R + } d= {a HK B H,K t), t R + },.3) where X = d Y means the two processes have the same finite dimensional distributions. Note that, when K =, B H,K is the ordinary fractional Brownian motion in R d. However, if K, B H,K does not have stationary increments. In fact, fractional Brownian motion is the only Gaussian self-similar process with stationary increments [see Samorodnitsky and Taqqu 994)]. 2

3 Russo and Tudor 26) have established some properties concerning the strong variations, local times and stochastic calculus of real-valued bifractional Brownian motion. An interesting property that deserves to be recalled is the fact that, when HK = 2, the quadratic variation of this family of stochastic processes which includes the standard Brownian motion when K = and HK = 2 ) on [, t] is equal to a constant times t. This is really remarkable since as far as we know these are the only Gaussian self-similar processes with this quadratic variation and besides the well-known case of Brownian motion, the other members of this family are not semimartingale. Taking into account this property, it is natural to ask if the bifractional Brownian motion B H,K with KH = 2 shares other properties with Brownian motion from the sample path regularity point of view). As it can be seen from the rest of the paper, the answer is often positive: for example, the bi-fbm with HK = 2 and Brownian motion satisfy the same forms of Chung s laws of the iterated logarithm and the Hölder conditions for their local times. The rest of this paper is organized as follows. In Section 2 we apply the Lamperti transformation to prove the strong local nondeterminism of B H,K. This property plays essential roles in proving most of our results. In Section 3 we derive small ball probability estimates and a stochastic integral representation for B H,K. Applying these results, we prove a version of the Chung s law of the iterated logarithm for bifractional Brownian motion. Section 4 is devoted to the study of local times of one-parameter bifractional Brownian motion and the corresponding N-parameter fields. In general, there are mainly two methods in studying local times of Gaussian processes: the Fourier analysis approach introduced by Berman and the Malliavin calculus approach. It is known that, the Fourier analysis approach combined with various properties of local nondeterminism yields strong regularity properties such as the oint continuity and sharp Hölder conditions for the local times [see Berman 973), Pitt 978), Geman and Horowitz 98), Xiao 997, 27)]; while the Malliavin calculus approach requires fewer conditions on the process and establishes regularity of the local times in the sense of Sobolev-Watanabe spaces [see Watanabe 984), Imkeller et al. 995), Eddahbi et al. 25)]. In this paper we make use of both approaches to obtain more comprehensive results on local times of bifractional Brownian motion and fields. Throughout this paper, an unspecified positive and finite constant is denoted by c, which may not be the same in each occurrence. More specific constants in Section i are numbered as c i,, c i,2, Strong local nondeterminism The following proposition is essential in this paper. From its proof, we see that the same conclusion holds for quite general self-similar Gaussian processes. Proposition 2. For all constants < a < b, B H,K is strongly locally ϕ-nondeterministic on I = [a, b] with ϕr) = r 2HK. That is, there exist positive constants c 2, and r such that for all t I and all < r min{t, r }, Var B H,K t) ) B H,K s) : s I, r s t r c 2, ϕr). 2.) 3

4 Proof We consider the centered stationary Gaussian process Y = {Y t), t R} defined through Lamperti s transformation [Lamperti 962)]: Y t) = e HK t B H,K e t ), for every t R. 2.2) The covariance function rt) := E Y )Y t) ) is given by rt) = [ e 2Ht + ) K e t ] 2HK 2 K e HKt = 2 K ehkt [ + e 2Ht ) K e t 2HK ]. 2.3) Hence rt) is an even function and, by 2.3) and the Taylor expansion, we verify that rt) = Oe βt ) as t, where β = min{h2 K), HK}. It follows that r ) L R). Also, by using 2.3) and the Taylor expansion again, we also have rt) 2 K t 2HK as t. 2.4) The stationary Gaussian process Y is sometimes called the Ornstein-Uhlenbeck process associated with B H,K [Note that it does not coincide with the solution of the fractional Langevin equation, see Cheridito et al. 23) for a proof in the case K = ]. By Bochner s theorem, Y has the following stochastic integral representation Y t) = e iλt W dλ), t R, 2.5) R where W is a complex Gaussian measure with control measure whose Fourier transform is r ). The measure is called the spectral measure of Y. Since r ) L R), the spectral measure of Y has a continuous density function fλ) which can be represented as the inverse Fourier transform of r ): fλ) = π We would like to prove f has the following asymptotic property rt) costλ) dt. 2.6) fλ) c 2,2 λ +2HK) as λ, 2.7) where c 2,2 > is an explicit constant depending only on HK. Note that 2.4) and the Tauberian theorem due to Pitman 968, Theorem 5) only imply λ fx) dx c λ 2HK as λ. Some extra Tauberian condition on f is usually needed if we wish to obtain 2.7) by using the Tauberian theorem; see Bingham et al. 987). In the following we give a direct proof of 2.7) by using 2.6) and an Abelian argument similar to that in the proof of Theorem of Pitman 968). Without loss of generality, we assume from now on that λ >. Applying integration-by-parts to 2.6), we get fλ) = πλ 4 r t) sintλ) dt 2.8)

5 with r t) = HK 2 K ehkt [ + e 2Ht ) K e 2Ht ) + e t ) e t ) 2HK ]. 2.9) We need to distinguish three cases: 2HK <, 2HK = and 2HK >. In the first case, it can be verified from 2.9) that r t) = Oe βt ) as t, hence r ) L R), and r t) 2 K HK t 2HK as t. 2.) We will also make use of the properties of higher order derivatives of rt). It is elementary to compute r t) and verify that, when 2HK <, we have r t) 2 K HK2HK ) t 2HK 2 as t 2.) and r t) = Oe βt ) as t which implies r ) L R). Moreover, we can show that r t) > for all t large enough and r t) is eventually monotone decreasing. The behavior of the derivatives of rt) is slightly different when 2HK =. 2.9) becomes and r t) = 2 K+ et/2 [ + e 2Ht ) K e 2Ht ) e t], 2.2) r t) = et/2 [ + e 2Ht ) K H )e 2Ht + e 4Ht) ] + e t. 2.3) 2K+2 Hence we have r ) = 2 K, r ) = H/2, and both r ) and r ) are in L R). When 2HK >, it can be shown that 2.) still holds, r t) = Oe βt ) as t, r t) > for all t large enough and r t) is eventually monotone decreasing. We omit the details. Now we proceed to prove 2.7). First we consider the case when < 2HK <. By a change of variable, we can write Hence fλ) = πλ 2 r t λ fλ) πλ 2 ) r /λ) = r t/λ) r /λ) ) sin t dt. 2.4) sin t dt. 2.5) Let p, ) be a fixed constant such that r t) > on [p, ). It follows from 2.) and the dominated convergence theorem that lim λ p r t/λ) r /λ) sin t dt = On the other hand, integration-by-parts yields p p r t/λ) sin t dt = r p/λ) cos p + λ 5 t 2HK sin t dt. 2.6) p r t/λ) cos t dt. 2.7)

6 Using the Riemann-Lebesgue lemma, we derive r t/λ) sin t dt r p/λ) cos p + r t/λ) cos t dt p λ p 2 r p/λ) 2.8). Hence we have lim sup λ p r t/λ) r /λ) sin t dt 2p2HK. 2.9) Combining 2.5), 2.6), 2.9) and letting p, we see that, when < 2HK <, 2.7) holds with c 2,2 = 2 K HKπ t 2HK sin t dt. Secondly we consider the case 2HK =. Since r t) is continuous and r ) = 2 K, 2.6) becomes lim λ p r t/λ) sin t dt = r ) p Using 2.7) and integration-by-parts again we derive p r t/λ) sin t dt = r p/λ) cos p + λ sin t dt = r ) cos p). 2.2) It follows from 2.2), 2.3) and the Riemann-Lebesgue lemma that lim λ p We see from the above and 2.4) that p r t/λ) cos t dt. 2.2) r t/λ) sin t dt = r ) cos p. 2.22) fλ) 2 K π λ 2 as λ, 2.23) This verifies that 4.) holds when 2HK =. Finally we consider the case < 2HK < 2. Note that 2.6) and 2.9) are not useful anymore and we need to modify the above argument. By using integration-by-parts to 2.8) we obtain fλ) = πλ 2 r t) costλ) dt. 2.24) Note that we have < 2HK 2 <. Hence r t) is integrable in the neighborhood of t =. Consequently the proof for this case is very similar to the case of < 2HK <. From 2.24) and 2.) we can verify that 2.7) holds as well and the constant c 2,2 is explicitly determined by H and K. Hence we have proved 2.7) in general. It follows from 2.7) and Lemma of Cuzick and DuPreez 982) [see also Xiao 27) for more general results] that Y = {Y t), t R} is strongly locally ϕ-nondeterministic on any interval J = [ T, T ] with ϕr) = r 2HK in the following sense: There exist positive constants δ and c 2,3 such that for all t [ T, T ] and all r, t δ), Var Y t) Y s) : s J, r s t δ ) c 2,3 ϕr). 2.25) 6

7 Now we prove the strong local nondeterminism of B H,K on I. To this end, note that B H,K t) = t HK Y log t) for all t >. We choose r = aδ. Then for all s, t I with r s t r we have r b log s log t δ. 2.26) Hence it follows from 2.25) and 2.26) that for all t [a, b] and r < r, Var B H,K t) ) B H,K s) : s I, r s t r = Var t HK Y log t) ) s HK Y log s) : s I, r s t r t 2HK Var Y log t) ) Y log s) : s I, r s t r 2.27) a 2HK Var Y log t) ) Y log s) : s I, r/b log s log t δ c 2,3 ϕr). This proves Proposition 2.. For use in next section, we list two properties of the spectral density fλ) of Y. They follow from 2.7) or, more generally, from 2.4) and the truncation inequalities in Loéve 977, p.29); see also Monrad and Rootzén 995). Lemma 2.2 There exist positive constants c 2,4 and c 2,5 such that for u >, λ 2 fλ) dλ c 2,4 u 2 HK) 2.28) λ <u and λ u fλ) dλ c 2,5 u 2HK. 2.29) We will also need the following lemma from Houdré and Villa 23). Lemma 2.3 There exist positive constants c 2,6 and c 2,7 such that for all s, t R +, we have [ ) 2 ] c 2,6 t s 2HK E B H,K t) B H,K s) c 2,7 t s 2HK. 2.3) 3 Chung s law of the iterated logarithm As applications of small ball probability estimates, Monrad and Rootzén 995), Xiao 997) and Li and Shao 2) established Chung-type laws of the iterated logarithm for fractional Brownian motion and other strongly locally nondeterministic Gaussian processes with stationary increments. However, there have been no results on Chung s LIL for self-similar Gaussian processes that do not have stationary increments [Recall that the class of self-similar Gaussian processes is large and fbm is the only such process with stationary increments]. 7

8 In this section, we prove the following Chung s law of the iterated logarithm for bifractional Brownian motion in R. It will be clear that our argument is applicable to a large class of self-similar Gaussian processes. Theorem 3. Let B H,K = {B H,K t), t R + } be a bifractional Brownian motion in R. Then there exists a positive and finite constant c 3, such that lim inf r max t [,r] B H,K t) r HK /log log/r)) HK = c 3, a.s. 3.) In order to prove Theorem 3., we need several preliminary results. Lemma 3.2 gives estimates on the small ball probability of B H,K. Lemma 3.2 There exist positive constants c 3,2 and c 3,3 such that for all t [, ] and x, ), exp c ) { 3,2 x /HK) P max B H,K t) B H,K t ) } x exp c ) 3,3 t [,] x /HK). 3.2) Proof By Proposition 2. and Lemma 2.3, we see that B H,K satisfies Conditions C) and C2) in Xiao 27). Hence this lemma follows from Theorem 3. in Xiao 27). Proposition 3.3 provides a zero-one law for ergodic self-similar processes, which complements the results of Takashima 989). In order to state it, we need to recall some definitions. Let X = {Xt), t R} be a separable, self-similar process with index κ. For any constant a >, the scaling transformation S κ,a of X is defined by S κ,a X)t) = a κ Xat), t R. 3.3) Note that X is κ-self-similar is equivalent to saying that for every a >, the process {S κ,a X)t), t R} has the same finite dimensional distributions as those of X. That is, for a κ-self-similar process X, a scaling transformation S κ,a preserves the distribution of X, and so the notion of ergodicity and mixing of S κ,a can be defined in the usual way, cf. Cornfeld et al. 982). Following Takashima 989), we say that a κ-self-similar process X = {Xt), t R} is ergodic or strong mixing) if for every a >, a, the scaling transformation S κ,a is ergodic or strong mixing, respectively). This, in turn, is equivalent to saying that the shift transformations for the corresponding stationary process Y = {Y t), t R} defined by Y t) = e κt Xe t ) are ergodic or strong mixing, respectively). Proposition 3.3 Let X = {Xt), t R} be a separable, self-similar process with index κ. We assume that X) = and X is ergodic. Then for any increasing function ψ : R + R +, we have PE κ,ψ ) = or, where E κ,ψ = { ω : there exists δ > such that } sup s t Xs) t κ ψt) for all < t δ. 3.4) 8

9 Proof We will prove that for every a >, the event E κ,ψ is invariant with respective to the transformation S κ,a. Then the conclusion follows from the ergodicity of X. Fix a constant a > and a. We consider two cases: i) a > and ii) a <. In the first case, since ψ is increasing, we have ψau) ψu) for all u >. Assume that a.s. there is a δ > such that sup s t Xs) t κ ψt) for all < t δ, 3.5) then sup a κ Xas) = a κ s t sup Xs) t κ ψt) for all < t δ/a. 3.6) s at This implies that E κ,ψ Sκ,a ) Eκ,ψ. By the self-similarity of X, these two events have the same probability, it follows that P { E κ,ψ Sκ,a )} Eκ,ψ =. This proves that Eκ,ψ is S κ,a -invariant and, hence, has probability or. In case ii), we have ψau) ψu) for all u > and the proof is similar to the above. If S κ,a X E κ,ψ, then we have X E κ,ψ. This implies Sκ,a ) Eκ,ψ Eκ,ψ and again E κ,ψ is S κ,a -invariant. This finishes the proof. By a result of Maruyama 949) on ergodicity and mixing properties of stationary Gaussian processes, we see that B H,K is mixing. Hence we have the following corollary of Proposition 3.3. Corollary 3.4 There exists a constant c 3,4 [, ] such that lim inf t + log log /t) HK t HK max B H,K s t s) = c 3,4, a.s. 3.7) Proof We take ψ c t) = c log log /t ) HK and define c3,4 = sup { c : P { } } E κ, ψc =. It can be verified that 3.7) follows from Proposition 3.3. It follows from Corollary 3.4 that Theorem 3. will be established if we show c 3,4, ). This is where Lemma 3.2 and the following lemma from Talagrand 995) are needed. Lemma 3.5 Let X = {Xt), t R} be a centered Gaussian process in R and let S R be a closed set equipped with the canonical metric defined by [ ds, t) = E Xs) Xt) ) ] 2 /2. Then there exists a positive constants c 3,5 such that for all u >, { P sup Xs) Xt) c 3,5 u + s, t S D ) } ) log Nd S, ε) dε exp u2 D 2, 3.8) where N d S, ε) denotes the smallest number of open d-balls of radius ε needed to cover S and where D = sup{ds, t) : s, t S} is the diameter of S. Now we proceed to prove Theorem 3.. 9

10 Proof of Theorem 3. We prove the lower bound first. For any integer n, let r n = e n. Let < γ < c 3,3 be a constant and consider the event { A n = max B H,K s) γ HK rn HK /log log /r n ) }. HK s r n Then the self-similarity of B H,K and Lemma 3.2 imply that P{A n } exp c ) 3,3 γ log n = n c 3,3 /γ. 3.9) Since n= P{A n} <, the Borel-Cantelli lemma implies max s [,rn ] B H,K s) lim inf n rn HK /log log/r n )) HK c 3,3 a.s. 3.) It follows from 3.) and a standard monotonicity argument that lim inf r max t [,r] B H,K t) r HK /log log/r)) HK c 3,6 a.s. 3.) The upper bound is a little more difficult to prove due to the dependence structure of B H,K. In order to create independence, we will make use of the following stochastic integral representation of B H,K : For every t >, B H,K t) = t HK e iλ log t W dλ). 3.2) This follows from the spectral representation 2.5) of Y and its connection with B H,K. For every integer n, we take R t n = n n and d n = n β, 3.3) where β > is a constant whose value will be determined later. It is sufficient to prove that there exists a finite constant c 3,7 such that max s [,tn ] B H,K s) lim inf n t HK n /log log/t n )) HK c 3,7 a.s. 3.4) and Let us define two Gaussian processes X n and X n by X n t) = t HK e iλ log t W dλ) 3.5) X n t) = t HK λ d n,d n] λ / d n,d n] e iλ log t W dλ), 3.6) respectively. Clearly B H,K t) = X n t) + X n t) for all t. It is important to note that the Gaussian processes X n n =, 2,...) are independent and, moreover, for every n, the processes X n and X n are independent as well. Denote hr) = r HK log log /r ) HK. We make the following two claims:

11 i). There is a constant γ > such that { P max Xn s) } γ HK ht n ) =. 3.7) n= s [,t n ] ii). For every ε >, n= { P max s [,t n ] Xn s) } > ε htn ) <. 3.8) Since the events in 3.7) are independent, we see that 3.4) follows from 3.7), 3.8) and a standard Borel-Cantelli argument. It remains to verify the claims i) and ii) above. By Lemma 3.2 and Anderson s inequality [see Anderson 955)], we have { P max Xn s) } { γ HK ht n ) P max B H,K s) } γ HK ht n ) s [,t n] s [,t n] exp c ) 3,2 γ logn log n) 3.9) = n log n ) c 3,2 /γ. Hence i) holds for γ c 3,2. In order to prove ii), we divide [, t n ] into p n + non-overlapping subintervals J n, = [a n,, a n, ], i =,,..., p n ) and then apply Lemma 3.5 to X n on each of J n,. Let β > be the constant in 3.3) and we take J n, = [, t n n β ]. After J n, has been defined, we take a n,+ = a n, + n β ). It can be verified that the number of such subintervals of [, t n ] satisfies the following bound: p n + c n β log n. 3.2) Moreover, for every, if s, t J n, and s < t, then we have t/s n β and this yields t t s s n β and log n s) β. 3.2) Lemma 2.3 implies that the canonical metric d for the process X n satisfies and d, s) c t HK n c t HK n n βhk and Some simple calculation yields D ds, t) c s t HK for all s, t > 3.22) n βhk for every s J n,. It follows that D := sup{ds, t); s, t J n, } log N d J n,, ε) dε N d J n,, ε) c t n n β. 3.23) ε/hk) t HK n n βhk = t HK n n βhk = c 3,8 t HK n n βhk. tn n log β ) ε /HK) dε log du u) 3.24)

12 It follows from Lemma 3.5 and 3.24) that { P max Xn s) } > ε htn ) s J n, exp c n 2βHK logn log n) ) 2HK ). 3.25) For every p n, we estimate the d-diameter of J n,. It follows from 3.6) that for any s, t J n, with s < t, E Xn s) X ) 2 n t) = t HK e iλ log t s HK e iλ log s 2 fλ) dλ λ d n + t HK e iλ log t s HK e iλ log s 2 fλ) dλ 3.26) := I + I 2. λ >d n The second term is easy to estimate: For all s, t J n,, I 2 4 t 2HK n fλ) dλ c 3,9 t 2HK n n 2βHK, 3.27) λ >d n where the last inequality follows from 2.29). For the first term I, we use the elementary inequality cos x x 2 to derive that for all s, t J n, with s < t, [ t I = HK s HK) 2 + 2t HK s HK cos λ log t ) )] fλ) dλ λ d n s 2HK t s ) 2HK c 3, t 2HK n n 2βHK, R s fλ) dλ + 2t 2HK log 2 t ) λ 2 fλ) dλ s λ d n where, in deriving the last inequality, we have used 3.2) and 2.28), respectively. It follows from 3.26), 3.27) and 3.28) that the d-diameter of J n, satisfies 3.28) D c 3, t HK n n βhk. 3.29) Hence, similar to 3.25), we use Lemma 3.5 and 3.29) to derive { P max Xn s) } n 2βHK > ε ht n ) exp c ) s J 2HK ). 3.3) n, logn log n) By combining 3.2), 3.25) and 3.3) we derive that for every ε >, { P max Xn s) } p n { > ε htn ) P max Xn s) } > ε htn ) s J n, n= s [,t n ] c <. n= = n β log n exp n= This proves 3.8) and hence the theorem. c n 2βHK ) ) 2HK logn log n) 3.3) 2

13 Remark 3.6 Let t [, ] be fixed and we consider the process X = {Xt), t R + } defined by Xt) = B H,K t + t ) B H,K t ). By applying Lemma 3.2 and modifying the proof of Theorem 3., one can show that c max t [,r] B H,K t + t ) B H,K t ) lim inf 3,2 r r HK /log log/r)) HK c 3,2 a.s., 3.32) where c 3,2 > is a constant depending on HK only. Corresponding to Lemma 3.2, we can also consider the small ball probability of B H,K under the Hölder-type norm. For α, ) and any function y C [, ]), we consider the α-hölder norm of y defined by y α = sup s,t [,],s t ys) yt) s t α. 3.33) The following proposition extends the results of Stolz 996) and Theorem 2. of Kuelbs, Li and Shao 995) to bifractional Brownian motion. Proposition 3.7 Let B H,K be a bifractional Brownian motion in R and α, HK). There exist positive constants c 3,3 and c 3,4 such that for all ε, ), exp c 3,3 ε /HK α)) { } P B H,K α ε exp c 3,4 ε /HK α)). 3.34) Proof It follows from Theorem 3.4 in Xiao 27). 4 Local times of bifractional Brownian motion This section is devoted to the study of the local times of the bi-fbm both in the one-parameter and multi-parameter cases. As we pointed out in the Introduction there are essentially two ways to prove the existence and regularity properties of local times for Gaussian processes: the first is related to the Fourier analysis and the local nondeterminism property; the second is based on the Malliavin calculus and Wiener-Itô chaos expansion. We will apply the Fourier analysis approach for the one-parameter case and the Malliavin calculus approach for the multiparameter case. 4. The one-parameter case Let B H,K = {B H,K t), t R + } be a bifractional Brownian motion with indices H and K in R d. For any closed interval I R + and for any x R d, the local time Lx, I) of B H,K is defined as the density of the occupation measure µ I defined by µ I A) = A B H,K s) ) ds, A BR d ). I 3

14 It can be shown [cf. Geman and Horowitz 98) Theorem 6.4] that the following occupation density formula hods: For every Borel function gt, x) on I R d, g t, B H,K t) ) dt = gt, x)lx, dt) dx. 4.) I Lemma 2.3 and Theorem 2.9 in Geman and Horowitz 98) imply that if /HK) > d then B H,K has a local time Lx, t) := Lx, [, t]), where x, t) R d [, ). In fact, more regularity properties of Lx, t) can be derived from Theorem 3.4 in Xiao 27) which we summarize in the following theorem. Besides interest in their own right, such results are also useful in studying the fractal properties of the sample paths of B H,K. Theorem 4. Let B H,K = {B H,K t), t R} be a bifractional Brownian motion with indices H and K in R d. If /HK) > d, then the following properties hold: R d I i) B H,K has a local time Lx, t) that is ointly continuous in x, t) almost surely. ii) [Local Hölder condition] For every B BR), let L B) = sup x R d Lx, B) be the maximum local time. Then there exists a positive constant c 4, such that for all t R +, lim sup r L Bt, r)) ϕ r) c 4, a.s. 4.2) iii) Here and in the sequel, Bt, r) = t r, t + r) and ϕ r) = r HKd log log /r) HKd. [Uniform Hölder condition] For every finite interval I R, there exists a positive finite constant c 4,2 such that lim sup r where ϕ 2 r) = r HKd log /r) HKd. L Bt, r)) sup c t I ϕ 2 r) 4,2 a.s., 4.3) Proof By Proposition 2. and Lemma 2.3, we see that the conditions of Theorem 3.4 in Xiao 27) are satisfied. Hence the results follow. The following states that the local Hölder condition for the maximum local time is sharp. Remark 4.2 By the definition of local times, we have that for every interval Q R +, Q = Lx, Q) dx L Q) max B H,K s) B H,K t) ) d. 4.4) B H,K Q) By taking Q = Bt, r) in 4.4) and using 3.32) in Remark 3.6, we derive the lower bound in the following L Bt, r)) c 4,3 lim sup c r ϕ r) 4,4 a.s., 4.5) 4 s,t Q

15 where c 4,3 > is a constant independent of t and the upper bound is given by 4.2). A similar lower bound for 4.3) could also be established by using 4.4), if one proves that for every interval I R +, lim inf r inf t I B H,K s) B H,K t) max s Bt,r) r Hk /log /r) HK c 4,5 a.s. 4.6) Theorem 4. can be applied to determine the Hausdorff dimension and Hausdorff measure of the level set Z x = {t R + : B H,K t) = x}, where x R d. See Berman 972), Monrad and Pitt 987) and Xiao 997, 27). In the following theorem we prove a uniform Hausdorff dimension result for the level sets of B H,K. Theorem 4.3 If /HK) > d, then with probability one, where dim H Proof dim H Z x = HKd for all x R d, 4.7) denotes Hausdorff dimension. It follows from Theorem 3.9 in Xiao 27) that with probability one, dim H Z x = HKd for all x O, 4.8) where O is the random open set defined by O = { } x R d : Lx, [s, t]) >. s,t Q; s<t Hence it only remains to show O = R d a.s. For this purpose, we consider the stationary Gaussian process Y = {Y t), t R} defined by Y t) = e HKt B H,K e t ), using the Lamperti transformation. Note that the component processes of Y are independent and, as shown in the proof of Proposition 2., they are strongly locally ϕ-nondeterministic with ϕr) = r 2HK. It follows from Theorem 3.4 in Xiao 27) that Y has a ointly continuous local time L Y x, t), where x, t) R d R. From the proof of Proposition 2., it can be verified that Y satisfies the conditions of Theorem 2 in Monrad and Pitt 987), it follows that almost surely for every y R d, there exists a finite interval J R such that L Y y, J) >. On the other hand, by using the occupation density formula 4.), we can verify that the local times of B H,K and Y are related by the following equation: For all x R d and finite interval I = [a, b] [, ), Lx, I) = e HK)s L Y e HKs x, ds). 4.9) [log a, log b] Hence, there exists a.s. a finite interval I such that L, I) >. The continuity of Lx, I) implies the a.s. existence of δ > such that Ly, I) > for all y R d with y δ. Observe that the scaling property of B H,K implies that for all constants c >, the scaled local time c HKd) Lx, ct) is a version of Lc HK x, t). It follows that a.s. for every x R d, Lx, J) > for some finite interval J [, ). 5

16 Since there is little knowledge on the explicit distribution of L, ), it is of interest to estimate the tail probability P{L, ) > x} as x. This problem has been considered by Kasahara et al. 999) for certain fractional Brownian motion and by Xiao 27) for a large class of Gaussian processes. Our next result is a consequence of Theorem 3.2 in Xiao 27). Theorem 4.4 Let B H,K = {B H,K t), t R} be a bifractional Brownian motion in R d with indices H and K. If /HK) > d, then for x > large enough, log P { L, ) > x } x HK, 4.) where ax) bx) means ax)/bx) is bounded from below and above by positive and finite constants for allx large enough. Proof By Proposition 2. and Lemma 2.3, we see that the conditions of Theorem 3.2 in Xiao 27) are satisfied. This proves 4.). Let us also note that the existence of the ointly continuous version of the local time and the self-similarity allow us to prove the following renormalization result. The case d = has been proved in Russo and Tudor 26). Proposition 4.5 If /HK) > d, then for any integrable function F : R d R, t HKd F B H,K u) ) du d) F L, ) as t, 4.) where F = R d F x) dx. [,t] Proof [,t] It holds that F B H,K u) ) du = t [,] F B H,K tv) ) dv = d t F t HK B H,K v) ) dv. 4.2) [,] By using the occupation density formula, we derive F B H,K u)) du = t F t HK x ) Lx, ) dx = t HKd [,t] R d R d F y) Lyt HK, ) dy. 4.3) Since the function y Ly, ) is almost surely continuous and bounded, the dominated convergence theorem implies that, as t, the last integral in 4.3) tends to F L, ) almost surely. This and 4.2) yield 4.). 6

17 4.2 Oscillation of bifractional Brownian motion The oscillations of certain classes of stochastic processes, especially Gaussian processes, in the measure space [, ], λ ), where λ is the Lebesgue measure in R, have been studied, among others, by Wschebor 992) and Azaïs and Wschebor 996). The following is an analogous result for bifractional Brownian motion. Proposition 4.6 Let B H,K be a bi-fbm in R with indices H, ) and K, ]. For every t [, ], let Z ε t) = BH,K t + ε) B H,K t) ε HK. Then the following statements hold: i) For every integer k, almost surely, Zε t) ) k dt Eρ k ) as ε, where ρ is a centered normal random variable with variance σ 2 = 2 K. ii) For every interval J [, ], almost surely, for every x R λ {t J : Z ε t) x} λ J) Pρ x) as ε. Proof Let us denote It is sufficient to prove that Y ε,k = Z ε t)) k dt. Var Y ε,k) ck) ε β for some ck) and β >. 4.4) Then the conclusions i) and ii) will follow as in Azaïs and Wschebor 996) by the means of a Borel-Cantelli argument. Note that Var Y ε,k) = Cov Z ε u) k, Z ε u) k) dudv. We will make use of the fact that for a centered Gaussian vector U, V ), Cov U k, V k) = cp, k) [ CovU, V ) ] p [ ] k p. VarU)VarV ) p k Since the random variable Z ε has clearly bounded variance [cf. Lemma 2.3], it suffices to show that for every p k, [ E Zε u)z ε v) )] p dudv c4,6 ε β 4.5) 7

18 We can write [ E Zε u)z ε v) )] p dvdu = u l u v<ε) [ E Zε u)z ε v) )] p dvdu u := A + B. Clearly A c ε, hence it suffices to bound the term B. Note that Since 2 R a b we have B cp, H, K) E Z ε u)z ε v) ) = ε 2HK u u ε l u v ε) [ E Zε u)z ε v) )] p dvdu v v ε 2 R a b dbda. a, b) = 2HK 2 K [ a 2H + b 2H) K 2 a 2H b 2H 2HK ) a b 2HK 2], := B + B 2. u ε + cp, H, K) [ ε 2HK u ε u u ε [ ε 2HK v v ε u u ε a 2H + b 2H) ] p K 2 a 2H b 2H dbda dvdu v v ε a b 2HK 2 dbda] p dvdu The term B 2 can be treated as in the fbm case [see Azaïs and Wschebor 996), Proposition 2.] and we get B 2 c ε β for some constant β >. Finally, since a 2HK + b 2HK a HK b HK, we can write u ε [ u v p B cp, H, K) a HK b dbda] HK dvdu = cp, H, K) u ε ε 2HK u ε v ε u HK u ε) HK ) p v HK v ε) HK ) p dvdu ε HK [ u HK u ε) HK ) p ] 2 c dvdu. ε HK A change of variable shows that B c ε 2 HK). Combining the above yields 4.5). Therefore, we have proved 4.4), and the proposition. ε HK The above result can be extended to obtain the almost sure weak approximation of the occupation measure of the bi-fbm B H,K by means of normalized number of crossing of Bε H,K, where Bε H,K represents the convolution of B H,K with an approximation of the identity Φ ε t) = ε Φ t ε) with Φ = l[,]. If g is a real function defined on an interval I, then the number of crossing of level u is N u g, I) = #{t I : gt) = u}, where #E denotes the cardinality of E. 8

19 Proposition 4.7 Almost surely for every continuous function f and for every bounded interval I R +, π ) /2 ε HK fu)n u Bε H,K, I) du fu)lu, I) du as ε. 2 Proof The arguments in Azaïs and Wschebor 996, Section 5) apply. Details are left to the reader. 4.3 The multi-parameter case For any given vectors H = H,..., H N ), ) N and K = K,..., K N ), ] N, an N, d)-bifractional Brownian sheet B H,K = {B H,K t), t R N + } is a centered Gaussian random field in R d with i.i.d. components whose covariance functions are given by ) N [ E B H,K s)b H,K ) ] t) = s 2H + t 2H K t s 2H K. 4.6) 2 K It follows from 4.6) that, similar to an N, d)-fractional Brownian sheet [cf. Xiao and Zhang 22), Ayache and Xiao 25)], B H,K is operator-self-similar in the sense that for all constants c >, { B H,K c A t), t R N} { d = c N B H,K t), t R N}, 4.7) where A = a i ) is the N N diagonal matrix with a ii = /H i K i ) for all i N and a i = if i, and X = d Y means that the two processes have the same finite dimensional distributions. However, it does not have convenient stochastic integral representations which have played essential rôles in the studies of fractional Brownian sheets. Nevertheless, we will prove that the sample path properties of B H,K are very similar to those of fractional Brownian sheets, and we can describe the anisotropic properties of B H,K in terms of the vectors H and K. We start with the following useful lemma. Lemma 4.8 For any ε >, there exist positive and finite constants c 4,7 and c 4,8 such that for all s, t [ε, ] N, N [ ) ] 2 N c 4,7 s t 2H K E B H,K s) B H,K t) c 4,8 s t 2H K, 4.8) and c 4,7 N ) N s t 2H K detcov B H,K s), B H,K t) c 4,8 s t 2H K. 4.9) Here and in the sequel, detcov denotes determinant of the covariance matrix. 9

20 Proof We will make use of the following easily verifiable fact: For any Gaussian random vector Z, Z 2 ), detcovz, Z 2 ) = VarZ )VarZ 2 Z ), 4.2) where VarZ ) and VarZ 2 Z ) denote the variance of Z and the conditional variance of Z 2, given Z, respectively. By 4.2) we see that for all s, t [ε, ] N, ) [ detcov B H,K s), B H,K t) = E B H,K s) 2] Var B H,K t) ) B H,K s) [ E B H,K s) 2] [ ) ] 2 4.2) E B H,K s) B H,K t). Since Var B H,K s) ) is bounded from above and below by positive and finite constants, it is sufficient to prove the upper bound in 4.8) and the lower bound in 4.9). Both of them are proven by induction. When N =, Lemma 2.3, Proposition 2. and 4.2) imply that both 4.8) and 4.9) hold. Next we show that, if the lemma holds for any B H,K with at most n parameters, then it holds for B H,K with n + parameters. We verify the upper bound in 4.8) first. For any s, t [ε, ] n+, let s = s,..., s n, t n+ ). Then we have [ ) ] [ 2 ) ] 2 E B H,K s) B H,K t) 2E B H,K s) B H,K s ) [ + 2E B H,K s ) B H,K t) ) ] ) For the first term, we note that whenever s,..., s n [ε, ] are fixed, B H,K is a rescaled) bifractional Brownian motion in s n+. Hence Lemma 2.3 implies the first term in the righthand side of 4.22) is bounded by c t n s n 2H n+k n+, where the constant c is independent of s,..., s n [ε, ]. On the other hand, when t n+ [ε, ] is fixed, B H,K is a rescaled) N, d)-bifractional Brownian sheet. Hence the induction hypothesis implies the second term in the right-hand side of 4.22) is bounded by c n t s 2H K. This and 4.22) together prove the upper bound in 4.8). Suppose the lower bound in 4.9) holds for any B H,K with at most n parameters. For N = n +, we have detcov B H,K s), B H,K t) ) n+ = t 2H K n+ 2 2K s 2H K [ t 2H 4.23) + s 2H ) ] K 2. t s 2H K By splitting the right-hand side or 4.23) and using the induction hypothesis, we derive 2

21 that detcov B H,K s), B H,K t) ) = n+ =2 t 2H K + 2 2K { n+ =2 s 2H K { s 2H K t 2H K 2 2K [ ] t 2H + s 2H ) 2 K t s 2H K t 2H K n+ c s t 2H K s 2H K n+ 2 2K =2 [ t 2H for all s, t [ε, ] N. This proves the lower bound in 4.9). [ t 2H + s 2H ) ] } K 2 t s 2H K + s 2H ) ] } K 2 t s 2H K 4.24) Applying Lemma 4.8, we can prove that many results in Xiao and Zhang 22), Ayache and Xiao 25) on sample path properties of fractional Brownian sheets, such as the Hausdorff dimensions of the range, graph and level sets and the existence local times, hold for B H,K as well. Theorem 4.9 is concerned with the existence of local times of B H,K. Theorem 4.9 Let B H,K = { B H,K t), t R N } + be an N, d)-bifractional Brownian sheet with parameters H, ) N and K, ] N. If d < N H K then for any N-dimensional closed interval I, ) N, B H,K has a local time Lx, I), x R d. Moreover, the local time admits the following L 2 -representation Lx, I) = 2π) d e i y,bh,k s) dsdy, x R d. 4.25) R d e i y,x I Remark 4. Although the existence of local times can also be proved by using the Malliavin calculus [see Proposition 4.5 below], we prefer to provide a Fourier analytic proof because: ) we can compare in this way the two methods and 2) the above theorem gives in addition the representation 4.25). Proof Without loss of generality, we may assume that I = [ε, ] N where ε >. Let λ N be the Lebesgue measure on I. We denote by µ the image measure of λ N under the mapping t B H,K t). That is, µa) = λ N {t I : B H,K t) A} for all Borel sets A R d. Then the Fourier transform of µ is µξ) = e i ξ, BH,K t) dt. 4.26) I 2

22 It follows from Fubini s theorem and 4.8) that ) E µξ) 2 dξ = E e i ξ, BH,K s) B H,K t) dξ dsdt R d I I R d = c [ I I E B H,K s) B H,K t) ) 2] d/2 dsdt c [ N s ] t 2H K d/2 dsdt. I I 4.27) The same argument in Xiao and Zhang 22, p. 24) shows that the last integral is finite whenever d < N H K. Hence, in this case, µ L 2 R d ) a.s. and Theorem 4.9 follows from the Plancherel theorem. Remark 4. Recently, Ayache, Wu and Xiao 27) have shown that fractional Brownian sheets have ointly continuous local times based on the sectorial local nondeterminism. It would be interesting to prove that B H,K is sectorially locally nondeterministic and to establish oint continuity and sharp Hölder conditions for the local times of B H,K. Now we consider the Hausdorff and packing dimensions of the image, graph and level set of B H,K. In order to state our theorems conveniently, we assume < H K... H N K N <. 4.28) We denote packing dimension by dim P ; see Falconer 99) for its definition and properties. The following theorems can be proved by using Lemma 4.8 and the same arguments as in Ayache and Xiao 25, Section 3). We leave the details to the interested reader. Theorem 4.2 With probability, dim H B H,K [, ] N) = dim P B H,K [, ] N) { = min d; N } H K 4.29) and dim H GrB H,K [, ] N) = dim P GrB H,K [, ] N) { N H = K if k H k K k H K + N k + H k K k )d if N k H K H K d, d < k H K, 4.3) where H K :=. Theorem 4.3 Let L x = {t, ) N : B H,K t) = x} be the level set of B H,K. The following statements hold: 22

23 i) ii) If N H < d, then for every x R d we have L x = a.s. If N H > d, then for every x R d and < ε <, with positive probability dim H Lx [ε, ] N) = dim P Lx [ε, ] N) { k } H k = min + N k H k d, k N H = k H k H + N k H k d, if k H d < k H. 4.3) 4.4 A Malliavin calculus approach Using the Malliavin calculus approach, we can study the local times of more general bifractional Brownian sheets. Consider the N d)-matrices where for any i =,..., d H = H,..., H d ) and K = K,..., K d ), H i = H i,,..., H i,n ) and K i = K i,,..., K i,n ) with H i,, ) and K i,, ] for every i =,..., d and =,..., N. We will say that the Gaussian field B H,K is an N, d)-bifractional Brownian sheet with indices H and K if ) B H,K t) = B H t),..., B H d t), t [, ) N and for every i =,..., d, the random field {B H i t), t R N + } is centered and has covariance function ) N E B H i,k i t)b H i,k i s) = R H i,k i s, t) = R H i,,k i, s, t ). As in Subsection 4., the local time Lx, t) t R N + and x R d ) of B H,K is defined as the density of the occupation measure µ t, defined by µ t A) = l A B H,K s) ) ds, A BR d ). [,t] Formally, we can write Lx, t) = [,t] δ x B H,K s) ) ds, 23

24 where δ x denotes the Dirac function and δ x Bs H,K ) is therefore a distribution in the Watanabe sense see Watanabe 984)). We need some notation. For x R, let p σ x) be the centered Gaussian kernel with variance σ >. Consider also the Gaussian kernel on R d given by p d σx) = d p σ x i ), x = x,..., x d ) R d. i= Denote by H n x) the n th Hermite polynomial defined by H x) = and for n, H n x) = )n n! x 2 exp 2 ) d n dx n exp We will make use of the following technical lemma. x2 2 ), x R. Lemma 4.4 For any H, ) and K, ], let us define the function Q H,K z) = RH,K, z) z HK, z, ] and Q H,K ) =. Then the function Q H,K takes values in [, ], Q H,K ) = and it is strictly increasing. Moreover, there exists a constant δ > such that for all z δ, ), Q H,K z)) n exp cδ, H, K)n z) 2HK). 4.32) Proof Clearly, the Cauchy-Schwarz inequality implies Q H,K z). Let us prove that the function Q H,K is strictly increasing. By computing the derivative Q H,K z) and multiplying this by z HK+, we observe that it is sufficient to show z) 2HK + z) + z 2H ) K z 2H ) > for all z, ). 4.33) If HK 2, since + z2h ) K + z, the left side in 4.33) can be minorized by + z 2H ) K z) 2HK + z 2H) and this is positive since z) 2HK. If HK > 2, we note that z) 2HK + z) + + z 2H ) K z 2H z) + z) + + z 2H ) K z 2 + z 2H ) K z 2 ) + + z 2H ) K z 2 + z 2H ) K. and this implies 4.33). Concerning the inequality 4.32), we note that Now by Taylor s formula Q H,K z) n = exp n log Q H,K z)) exp n Q H,K z))). + z 2H ) K z HK 2 K + ch, K, δ) z) 2 24

25 and therefore Q H,K z) + ch, K, δ) z) 2 z)2hk 2K + ch, K, δ) z) 2HK δ 2 2HK 2 K z)2hk. The conclusion follows as in the proof of Lemma 2 in Eddahbi et al. 25), since Q H,K z) 2 K z)2hk ch, K, δ)) for any z δ, ) with δ close to zero and with ch, K, δ) tending to zero as δ. The following proposition gives a chaotic expansion of the local time of the N, d)- bifractional Brownian sheet. The stochastic integral I n h) appeared below is the multiple Wiener-Itô integral of order n of the function h of nn variables with respect to an N, ) bifractional Brownian motion with parameters H = H,..., H N ) and K = K,..., K N ). Recall that such integrals can be constructed in general on a Gaussian space [see, for example, Maor 98), or Nualart 995)]. We will only need the following isometry formula: ) E I n l n [,t] )I ml m [,s] ) = n! R H,K t, s) n l n=m) = n! for all s, t R N +. N R H,K t, s ) ) n ln=m) 4.34) Proposition 4.5 For any x R d and t, ) N, the local times Lx, t) admits the following chaotic expansion Lx, t) = n,...,n d [, t] i= d p s 2H iki x i ) s n ih i K i xi ) H ni I s H n i i l [,s] ) n i ) ds, 4.35) i where s = s s N and s H ik i = N sh i,k i,. The integrals In i i denotes the multiple Itô stochastic integrals with respect to the independent N-parameter bifractional Brownian motion B H i,k i. Moreover, if N H > d, where H K = max{h i, : i =,..., d} and K = max{k i, : i =,..., d}, then Lx, t) is a random variable in L 2 Ω). Proof The chaotic expression 4.35) can be obtained similarly as in Eddahbi et al. 25) or Russo and Tudor 26). It is based on the approximation of the Dirac delta function by Gaussian kernels with variance converging to zero. Let us evaluate the L 2 Ω) norm of Lx, t). By the independence of components and the isometry of multiple stochastic integrals, we obtain Lx, t) 2 2 = d du dv β ni u)β ni v)r H i,k i u, v) n i, 4.36) m n + +n d =m [,t] [,t] i= 25

26 where β ni u) = p s 2H ik i x i ) s n ih i K i By Propositions 3 and 6 in Imkeller et al. 995)], we have the bound β ni u)β ni v) c 4,9 H ni xi s H ik i ). 995) [see also Lemma in Eddahbi et al. n i ) 8β 6 u n ih i K iv n i H i K i 4.37) for any β [ 4, 2 ). Using the inequality 4.37), we derive from 4.36) that Lx, t) 2 2 is at most c m = c m n + +n d =m = c 4, t 2 m d n + +n d =m i= n i ) 8β 6 d i= n + +n d =m ) n i ) 8β 6 d i= du [,t] ) N n i ) 8β 6 t ) N [,u] dv d N i= R H i,,k i, u, v ) n i u v ) n ih i, K i, d u du Q Hi,,K i, z) n i i= i= d ) Q Hi,,K i, z) n i dz, ) dz 4.38) where we have used the change of variables u = u and v = z u. Using the above lemma and as in the proof of Lemma 2 in Eddahbi et al. 25), we can prove the bound d Q Hi,,K i, z) n i )dz c 4, m 2H K. 4.39) i= Here c 4, = c 4, H, K) depends on H, K. Finally, 4.39) implies that N Lx, t) 2 2 c 4,2 m c 4,3 m m N ) m 2H K n + +n d =m d i= n i ) 8β 6 2H K +d 8β 6 ), 4.4) where c 4,2 and c 4,3 depend on H, K and t only. The last series in 4.4) converges if ) N 2H K > d 8β ). 4.4) 6 To conclude, observe that by choosing β close to 2, N condition 4.4). H K > d implies the required 26

27 We recall that a random variable F = n I nf n ) belongs to the Watanabe space D α,2 if F 2 α,2 := n + m) α I n f n ) 2 2 <. Corollary 4.6 For every t, ) N and x R d, the local time Lx, t) of the N, d)- bifractional Brownian sheet B H,K belongs to the Watanabe space D α,2 for every < α < d 2. N 2H K Proof This is a consequence of the proof of Proposition 4.5. Using the computation contained there, we obtain for any β [ 4, 2 ), Lx, t) 2 α,2 c 4,4 H, K, d, t) m + m) α 8β d ) N 6 m 2H K, which is convergent if α < N to 2, we get the conclusion. 2H K d 8β 6 ) N 2H. Choosing β close K Acknowledgment This work was initiated while both authors were attending the Second Conference on Self-similarity and Applications held during June 2 24, 25, at INSA Toulouse, France. We thank the organizers, especially Professor Serge Cohen, for their invitation and hospitality. We also thank Professors N. H. Bingham and A. Inoue for stimulating discussions. References [] R. Addie, P. Mannersalo and I. Norros 22), Performance formulae for queues with Gaussian input. European Trans. Telecommunications 33), [2] T. W. Anderson 955), The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6, [3] V. V. Anh, J. M. Angulo and M. D. Ruiz-Medina 999), Possible long-range dependence in fractional random fields. J. Statist. Plann. Inference 8, 95. [4] A. Ayache, D. Wu and Y. Xiao 27), Joint continuity of the local times of fractional Brownian sheets. Ann. Inst. H. Poincaré Probab. Statist., to appear. [5] A. Ayache and Y. Xiao 25), Asymptotic properties and Hausdorff dimensions of fractional Brownian sheets. J. Fourier Anal. Appl., [6] J. M. Azaïs and M. Wschebor 996), Almost sure oscillation of certain random processes. Bernoulli 2, [7] A. Benassi, P. Bertrand et J. Istas 2), Identification of the Hurst exponent of a step multifractional Brownian motion. Statist. Inference Stoch. Process. 3,. 27

28 [8] D. A. Benson, M. M. Meerschaert and B. Baeumer 26), Aquifer operator-scaling and the effect on solute mixing and dispersion. Water Resour. Res. 42, W45. [9] S. M. Berman 972), Gaussian sample function: uniform dimension and Hölder conditions nowhere. Nagoya Math. J. 46, [] S. M. Berman 973), Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23, [] N. H. Bingham, C. M. Goldie and J. L. Teugels 987), Regular Variation. Cambridge University Press, Cambridge. [2] A. Bonami and A. Estrade 23), Anisotropic analysis of some Gaussian models. J. Fourier Anal. Appl. 9, [3] P. Cheridito 24), Gaussian moving averages, semimartingales and option pricing. Stoch. Process. Appl. 9, [4] P. Cheridito, H. Kawaguchi and M. Maeima 23), Fractional Ornstein-Uhlenbeck processes. Electron. J. Probab. 8, paper 3, pp. 4. [5] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai 982), Ergodic Theory. Springer, New York. [6] J. Cuzick and J. DuPreez 982), Joint continuity of Gaussian local times. Ann. Probab., [7] M. Eddahbi, R. Lacayo, J. L. Solé, C. A. Tudor, J. Vives 25), Regularity of the local time for the d dimensional fractional Brownian motion with N-parameters. Stoch. Anal. Appl. 23, [8] K. J. Falconer 99), Fractal Geometry Mathematical Foundations and Applications. Wiley & Sons, Chichester. [9] D. Geman and J. Horowitz 98), Occupation densities. Ann. Probab. 8, 67. [2] C. Houdré and J. Villa 23), An example of infinite dimensional quasi-helix. In: Stochastic models Mexico City, 22), pp.95 2, Contemp. Math., 336, Amer. Math. Soc., Providence, RI. [2] P. Imkeller, V. Perez Abreu and J. Vives 995), Chaos expansion of double intersection local time of Brownian motion in R d and renormalization. Stoch. Process. Appl. 56, 34. [22] Y. Kasahara, N. Kôno and T. Ogawa 999), On tail probability of local times of Gaussian processes. Stoch. Process. Appl. 82, 5 2. [23] J. Kuelbs, W. V. Li and Q.-M. Shao 995), Small ball probabilities for Gaussian processes with stationary increments under Hölder norms. J. Theoret. Probab. 8, [24] J. Lamperti 962), Semi-stable stochastic processes. Trans. Amer. Math. Soc. 4, [25] W. V. Li and Q.-M. Shao 2), Gaussian processes: inequalities, small ball probabilities and applications. In: Stochastic Processes: Theory and Methods. Handbook of Statistics, 9, C. R. Rao and D. Shanbhag, editors), pp , North-Holland. [26] L. Loéve 977), Probability Theory I. Springer, New York. [27] P. Maor 98), Multiple Wiene-Itô Integrals. Lecture Notes in Math. 849, Sringer-Verlag, Berlin. 28