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Journal of Inequalities and Applications This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Necessary and sufficient condition for the smoothness of intersection local time of subfractional Brownian motions Journal of Inequalities and Applications 2011, 2011:139 doi:10.1186/1029-242X-2011-139 Guangjun Shen (guangjunshen@yahoo.com.cn) ISSN 1029-242X Article type Research Submission date 6 September 2011 Acceptance date 19 December 2011 Publication date 19 December 2011 Article URL http://www.journalofinequalitiesandapplications.com/content/2011/1/ This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Journal of Inequalities and Applications go to http://www.journalofinequalitiesandapplications.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 Shen ; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Necessary and suﬃcient condition for the smoothness of intersection local time of subfractional Brownian motions Guangjun Shen Department of Mathematics, Anhui Normal University, Wuhu 241000, China Email address: guangjunshen@yahoo.com.cn Abstract Let S H and S H be two independent d-dimensional sub- fractional Brownian motions with indices H ∈ (0, 1). Assume d ≥ 2, we investigate the intersection local time of subfractional Brownian motions TT H H = δ St − Ss dsdt, T > 0, T 0 0 where δ denotes the Dirac delta function at zero. By elementary inequalities, we show that T exists in L2 if and only if Hd < 2 and it is smooth in the sense of the Meyer-Watanabe if and only if 2 H < d+2 . As a related problem, we give also the regularity of the intersection local time process. 2010 Mathematics Subject Classiﬁcation: 60G15; 60F25; 60G18; 60J55. Keywords: subfractional Brownian motion; intersection local time; Chaos expansion. 1. Introduction The intersection properties of Brownian motion paths have been in- vestigated since the forties (see [1]), and since then, a large number of results on intersection local times of Brownian motion have been accumulated (see Wolpert [2], Geman et al. [3], Imkeller et al. [4], de Faria et al. [5], Albeverio et al. [6] and the references therein). The intersection local time of independent fractional Brownian motions has been studied by Chen and Yan [7], Nualart et al. [8], Rosen [9], Wu and Xiao [10] and the references therein. As for applications in physics, the Edwards, model of long polymer molecules by Brownian motion paths uses the intersection local time to model the ‘excluded volume’ eﬀect: diﬀerent parts of the molecule should not be located at the same point in space, while Symanzik [11], Wolpert [12] introduced the intersection local time as a tool in constructive quantum ﬁeld theory. 1
2 Intersection functionals of independent Brownian motions are used in models handling diﬀerent types of polymers (see, e.g., Stoll [13]). They also occur in models of quantum ﬁelds (see, e.g., Albeverio [14]). As an extension of Brownian motion, recently, Bojdecki et al. [15] introduced and studied a rather special class of self-similar Gaussian processes, which preserves many properties of the fractional Brown- ian motion. This process arises from occupation time ﬂuctuations of branching particle systems with Poisson initial condition. This process is called the subfractional Brownian motion. The so-called subfrac- tional Brownian motion (sub-fBm in short) with index H ∈ (0, 1) is a mean zero Gaussian process S H = {StH , t ≥ 0} with S0 = 0 and H 1 CH (s, t) := E StH Ss = s2H + t2H − (s + t)2H + |t − s|2H (1.1) H 2 1 H for all s, t ≥ 0. For H = 2 , S coincides with the Brownian motion B . S H is neither a semimartingale nor a Markov process unless H = 1/2, so many of the powerful techniques from stochastic analysis are not available when dealing with S H . The sub-fBm has self-similarity and long-range dependence and satisﬁes the following estimates: 2 [(2 − 22H −1 ) ∧ 1](t − s)2H ≤ E StH − Ss H ≤ [(2 − 22H −1 ) ∨ 1](t − s)2H . (1.2) Thus, Kolmogorov’s continuity criterion implies that sub-fBm is H¨ldero continuous of order γ for any γ < H. But its increments are not sta- tionary. More works for sub-fBm can be found in Bardina and Bas- compte [16], Bojdecki et al. [17–19], Shen et al. [20–22], Tudor [23] and Yan et al. [24, 25]. In the present paper, we consider the intersection local time of two independent sub-fBms on Rd , d ≥ 2, with the same indices H ∈ (0, 1). This means that we have two d-dimensional independent centered Gauss- ian processes S H = {StH , t ≥ 0} and S H = {StH , t ≥ 0} with covariance structure given by E StH,i Ss = E StH,i Ss H,j H,j = δi,j CH (s, t), where i, j = 1, . . . , d, s, t ≥ 0. The intersection local time can be for- mally deﬁned as follows, for every T > 0, T T δ StH − Ss dsdt, H = (1.3) T 0 0 where δ (·) denotes the Dirac delta function. It is a measure of the amount of time that the trajectories of the two processes, S H and S H ,
3 intersect on the time interval [0, T ]. As we pointed out, this deﬁnition is only formal. In order to give a rigorous meaning to T , we approximate the Dirac delta function by the heat kernel |x|2 d pε (x) = (2πε)− 2 e− x ∈ Rd . , 2ε Then, we can consider the following family of random variables indexed by ε > 0 T T pε (StH − Ss )dsdt, H = (1.4) ε,T 0 0 that we will call the approximated intersection local time of S H and S H . An interesting question is to study the behavior of ε,T as ε tends to zero. For H = 1 , the process S H and S H are Brownian motions. The inter- 2 section local time of independent Brownian motions has been studied by several authors (see Wolpert [2], Geman et al. [3] and the references 1 therein). In the general case, that is H = 2 , only the collision local time has been studied by Yan and Shen [24]. Because of interesting properties of sub-fBm, such as short-/long-range dependence and self- similarity, it can be widely used in a variety of areas such as signal processing and telecommunications( see Doukhan et al. [26]). There- fore, it seems interesting to study the so-called intersection local time for sub-fBms, a rather special class of self-similar Gaussian processes. The aim of this paper is to prove the existence, smoothness, regu- larity of the intersection local time of S H and S H , for H = 1 and 2 d ≥ 2. It is organized as follows. In Section 2, we recall some facts for the chaos expansion. In Section 3, we study the existence of the intersection local time. In Section 4, we show that the intersection local time is smooth in the sense of the Meyer-Watanabe if and only if 2 H < d+2 . In Section 5, the regularity of the intersection local time is also considered. 2. Preliminaries In this section, ﬁrstly, we recall the chaos expansion, which is an orthogonal decomposition of L2 (Ω, P ). We refer to Meyer [27] and Nualart [28] and Hu [29] and the references therein for more details. Let X = {Xt , t ∈ [0, T ]} be a d−dimensional Gaussian process deﬁned on the probability space (Ω, F , P ) with mean zero. If pn (x1 , . . . , xk ) is a polynomial of degree n of k variables x1 , . . . , xk , then we call pn (Xti1 , . . . , Xtik ) a polynomial functional of X with t1 , . . . , tk ∈ [0, T ] 1 k and 1 ≤ i1 , . . . , ik ≤ d. Let Pn be the completion with respect to the
4 L2 (Ω, P ) norm of the set {pm (Xti1 , . . . , Xtik ) : 0 ≤ m ≤ n}. Clearly, Pn 1 k is a subspace of L2 (Ω, P ). If Cn denotes the orthogonal complement of Pn−1 in Pn , then L2 (Ω, P ) is actually the direct sum of Cn , i.e., ∞ 2 L (Ω, P ) = Cn . (2.1) n=0 For F ∈ L2 (Ω, P ), we then see that there exists Fn ∈ Cn , n = 0, 1, 2, . . . , such that ∞ F= Fn , (2.2) n=0 This decomposition is called the chaos expansion of F . Fn is called the n-th chaos of F . Clearly, we have ∞ E (|F |2 ) = E (|Fn |2 ). (2.3) n=0 As in the Malliavin calculus, we introduce the space of “smooth” functionals in the sense of Meyer and Watanabe (see Watanabe [30]): ∞ ∞ 2 nE (|Fn |2 ) < ∞}, U := {F ∈ L (Ω, P ) : F = Fn and n=0 n=0 and F ∈ L2 (Ω, P ) is said to be smooth if F ∈ U . Now, for F ∈ L2 (Ω, P ), we deﬁne an operator Υu with u ∈ [0, 1] by ∞ un Fn . Υu F := (2.4) n=0 d (||Θ(u)||2 ), Set Θ(u) := Υ√u F . Then, Θ(1) = F. Deﬁne ΦΘ (u) := du where ||F ||2 := E (|F |2 ) for F ∈ L2 (Ω, P ). We have ∞ nun−1 E (|Fn |2 ). ΦΘ (u) = (2.5) n=1 ∞ Note that ||Θ(u)||2 = E (|Θ(u)|2 ) = E (un |Fn |2 ). n=1 Proposition 1. Let F ∈ L2 (Ω, P ). Then, F ∈ U if and only if ΦΘ (1) < ∞. Now consider two d-dimensional independent sub-fBms S H and S H with indices H ∈ (0, 1). Let Hn (x), x ∈ R be the Hermite polynomials of degree n. That is, 1 x2 ∂ n − x2 Hn (x) = (−1)n e 2 e 2. (2.6) ∂xn n!
5 Then, ∞ 2 tx− t2 tn Hn (x) e = (2.7) n=0 for all t ∈ C and x ∈ R, which deduces 1 exp(iu ξ , StH − Ss + u2 |ξ |2 Var(StH,1 − Ss 1 )) H H, 2 ∞ ξ , StH − Ss H (iu)n σ n (t, s, ξ )Hn = , σ (t, s, ξ ) n=0 where σ (t, s, ξ ) = Var(StH,1 − Ss 1 )|ξ |2 for ξ ∈ Rd . Because of the H, orthogonality of {Hn (x), x ∈ R}n∈Z+ , we will get from (2.2) that ξ , StH − Ss H (iu)n σ n (t, s, ξ )Hn σ (t, s, ξ ) is the n-th chaos of 1 exp iu ξ , StH − Ss + u2 |ξ |2 Var StH,1 − Ss 1 H H, 2 for all t, s ≥ 0. 3. Existence of the intersection local time The aim of this section is to prove the existence of the intersection local time of S H and S H , for an H = 1 and d ≥ 2. We have obtained 2 the following result. converges in L2 (Ω). The Theorem 2. (i) If Hd < 2, then the ε,T limit is denoted by T (ii) If Hd ≥ 2, then lim E ( ε,T ) = +∞, ε→0 and lim Var( ε,T ) = +∞ . ε→0 1 Note that if {St2 }t≥0 is a planar Brownian motion, then T T 1/2 − Ss /2 dsdt, 1 = pε St ε 0 0 diverges almost sure, when ε tends to zero. Varadhan, in [31], proved that the renormalized self-intersection local time deﬁned as limε→0 ( ε −
6 E ε ) exists in L2 (Ω). Condition (ii) implies that Varadhan renormal- ization does not converge in this case. For Hd ≥ 2, according to Theorem 2, ε,T does not converge in L (Ω), and therefore, T , the intersection local time of S H and S H , 2 does not exist. Using the following classical equality 1 1 |x|2 |ξ|2 − ei ξ,x e−ε p ε ( x) = de = dξ, 2ε 2 (2π )d (2πε) 2 Rd we have T T p (StH − Ss )dsdt H = ε,T 0 0 (3.1) T T 1 |ξ|2 H −S H ei ξ,St −ε 2 = ·e dξ dsdt. s (2π )d 0 0 Rd Since ξ , StH − Ss ∼ N (0, |ξ |2 (2 − 22H −1 )(t2H + s2H )), so H 2 2H −1 )(t2H +s2H )] |ξ| H −S H E [ei ξ,St ] = e−[(2−2 . s 2 Therefore, T T 1 |ξ|2 H −S H E [ei ξ,St ] · e−ε E ( ε,T ) = dξ dsdt s 2 (2π )d 0 0 Rd T T 1 2 2H −1 )(t2H +s2H )] |ξ| e−[ε+(2−2 (3.2) = dξ dsdt 2 (2π )d 0 0 Rd T T 1 d [ε + (2 − 22H −1 )(t2H + s2H )]− 2 dsdt, = d (2π ) 2 0 0 where we have used the fact d 2π |ξ|2 2 −[ε+(2−22H −1 )(t2H +s2H )] e dξ = . 2 2H −1 )(t2H + s2H ) ε + (2 − 2 Rd
7 We also have 1 H H H H E ei ξ ,St −Ss +i η ,Su −Sv 2 E( ε,T ) = (2π )2d (3.3) [0,T ]4 R2d ε(|ξ|2 +|η |2 ) − ×e dξdη dsdtdudv. 2 Let we introduce some notations that will be used throughout this paper, λs,t = Var(StH,1 − Ss 2 ) = (2 − 22H −1 )(t2H + s2H ), H, ρu,v = Var(Sv 1 − Su 2 ) = (2 − 22H −1 )(u2H + v 2H ), H, H, and µs,t,u,v = Cov StH,1 − Ss 2 , Sv 1 − Su 2 H, H, H, 1 = s2H + t2H + u2H + v 2H − [(t + v )2H + |t − v |2H + (s + u)2H + |s − u|2H ], 2 where S H,1 and S H,2 are independent one dimensional sub-fBms with indices H . Using the above notations, we can write for any ε > 0 1 1 2 (λs,t + ε)|ξ |2 + (ρu,v + ε)|η |2 + 2µs,t,u,v ξ , η E( ε,T ) = exp − (2π )2d 2 [0,T ]4 R2d × dξ dη dsdtdudv 1 −d (λs,t + ε)(ρu,v + ε) − µ2 = dsdtdudv. 2 s,t,u,v (2π )d [0,T ]4 (3.4) In order to prove the Theorem 2, we need some auxiliary lemmas. Without loss of generality, we may assume v ≤ t, u ≤ s and v = xt, u = ys with x, y ∈ [0, 1]. Then, we can rewrite ρu,v and µs,t,u,v as following. ρu,v = (2 − 22H −1 )(x2H t2H + y 2H s2H ), 1 µs,t,u,v = t2H 1 + x2H − [(1 + x)2H + (1 − x)2H ] 2 (3.5) 1 + s2H 1 + y 2H − [(1 + y )2H + (1 − y )2H ] . 2 It follows that λs,t ρu,v − µ2 4H f (x) + s4H f (y ) + t2H s2H g (x, y ), s,t,u,v = t (3.6)
8 where 2 1 1 2H −1 2 2H 2H − (1 + x)2H − (1 − x)2H f (x) := (2 − 2 )x − 1+x , 2 2 and g (x, y ) =(2 − 22H −1 )2 x2H + y 2H 1 1 − 2 1 + x2H − (1 + x)2H − (1 − x)2H (3.7) 2 2 1 1 × 1 + y 2H − (1 + y )2H − (1 − y )2H . 2 2 For simplicity throughout this paper, we assume that the notation F G means that there are positive constants c1 and c2 so that c1 G(x) ≤ F (x) ≤ c2 G(x) in the common domain of deﬁnition for F and G. For a, b ∈ R, a ∧ b := min{a, b} and a ∨ b := max{a, b}. By Lemma 4.2 of Yan and Shen [24], we get Lemma 3. Let f (x) and g (x, y ) be deﬁned as above and let 0 < H < 1. Then, we have f (x) x2H (1 − x)2H , (3.8) and g (x, y ) x2H (1 − y )2H + y 2H (1 − x)2H (3.9) for all x, y ∈ [0, 1]. Lemma 4. Let −d λs,t ρu,v − µ2 AT := dsdtdudv. 2 s,t,u,v [0,T ]4 Then, AT is ﬁnite if and only if Hd < 2. Proof. It is easily to prove the necessary condition. In fact, we can ﬁnd ε > 0 such that Dε ⊂ [0, T ]4 , where Dε ≡ (s, t, u, v ) ∈ R4 : s2 + t2 + u2 + v 2 ≤ ε2 . + We make a change to spherical coordinates as following  s = r cos ϕ1 ,   t = r sin ϕ cos ϕ , 1 2 (3.10) u = r sin ϕ1 sin ϕ2 cos ϕ3 ,    v = r sin ϕ1 sin ϕ2 sin ϕ3 .
9 where 0 ≤ r ≤ ε, 0 ≤ ϕ1 , ϕ2 ≤ π, 0 ≤ ϕ3 ≤ 2π , ∂ (s, t, u, v ) = r3 sin2 ϕ1 sin ϕ2 . J= ∂ (r, ϕ1 , ϕ2 , ϕ3 ) As λs,t ρu,v − µ2 2 4H s,t,u,v is always positive, and λs,t ρu,v − µs,t,u,v = r φ(θ), we have ε −d (λs,t ρu,v − µ2 r3−2Hd AT ≥ s,t,u,v ) dsdtdudv = φ(θ)dθ, (3.11) 2 0 Dε Θ where the integral in r is convergent if and only if 3 − 2Hd > −1 i.e., Hd < 2 and the angular integral is diﬀerent from zero thanks to the positivity of the integrand. Therefore, Hd ≥ 2 implies that AT = +∞. Now, we turn to the proof of suﬃcient condition. Suppose that Hd < 2. By symmetry, we have d (λs,t ρu,v − µ2 −2 AT = 4 s,t,u,v ) dsdtdudv, Υ where Υ = {(u, v, s, t) : 0 < u < s ≤ T, 0 < v < t ≤ T }. By Lemma 3, we get λs,t ρu,v − µ2 4H f (x) + s4H f (y ) + t2H s2H g (x, y ) s,t,u,v = t t4H x2H (1 − x)2H + s4H y 2H (1 − y )2H + t2H s2H (x2H (1 − y )2H + y 2H (1 − x)2H ) = [x2H t2H + y 2H s2H ][(1 − x)2H t2H + (1 − y )2H s2H ] = (v 2H + u2H )[(t − v )2H + (s − u)2H ]. (3.12) These deduce for all H ∈ (0, 1) and T > 0, T t T s H H −d/2 (uH (s − u)H )−d/2 du ΛT ≤ CH dt (v (t − v ) ) dv ds 0 0 0 0  2 T 1 Hd Hd = CH  dx < ∞. t1−Hd dt x− (1 − x)− 2 2 0 0 Proof of Theorem 2. Suppose Hd < 2, we have 1 −d ((λs,t + ε)(ρu,v + η ) − µ2 E ( ε,T · η,T ) = s,t,u,v ) 2 dsdtdudv. d (2π ) [0,T ]4
10 Consequently, a necessary and suﬃcient condition for the convergence in L2 (Ω) of ε,T is that d (λs,t ρu,v − µ2 −2 s,t,u,v ) dsdtdudv < ∞. [0,T ]4 This is true due to Lemma 4. If Hd ≥ 2, then from (3.2) and using monotone convergence theorem T T 1 d (s2H + t2H )− 2 dsdt. lim E ( ε,T ) = (2π (2 − 22H −1 ))d/2 ε→0 0 0 Making a polar change of coordinates x = r cos θ, y = r sin θ, where 0 ≤ r ≤ T, 0 ≤ θ ≤ π , 2 T T d (s2H + t2H )− 2 dsdt 0 0 π T 2 d r1−Hd (cos2H θ + sin2H θ)− 2 drdθ, = 0 0 and this integral is divergent if Hd ≥ 2. By the expression (3.2) and (3.4), we have 2 2 lim Var( ε,T ) = lim[E ( ε,T ) − (E ε,T ) ] ε→0 ε→0 1 −d d λs,t ρu,v − µ2 − (λs,t ρu,v )− 2 dv dudsdt. = 2 s,t,u,v (2π )d [0,T ]4 Making a change of variables to spherical coordinates as (3.10), if Hd ≥ 2, we have lim V ar( ε,T ) = +∞. ε→0
11 In fact, as the integrand is always positive, we obtain −d d λs,t ρu,v − µ2 − (λs,t ρu,v )− 2 dv dudsdt 2 s,t,u,v [0,T ]4 −d d λs,t ρu,v − µ2 − (λs,t ρu,v )− 2 dv dudsdt ≥ 2 s,t,u,v D r3−2Hd dr = ψ (θ)dθ, 0 Θ where the integral in r is convergent if and only if Hd < 2, and the angular integral is diﬀerent from zero thanks to the positivity of the integrand. Therefore, Hd ≥ 2 implies that lim Var( ε,T ) = +∞ . ε→0 This completes the proof of Theorem 2. 4. Smoothness of the intersection local time In this section, we consider the smoothness of the intersection local time. Our main object is to explain and prove the following theorem. The idea is due to An and Yan [32] and Chen and Yan [7]. Theorem 5. Let T be the intersection local time of two independent d-dimensional sub-fBms S H and S H with indices H ∈ (0, 1). Then, T ∈ U if and only if 2 H< . d+2 Recall that λs,t = (2 − 22H −1 )(t2H + s2H ), ρu,v = (2 − 22H −1 )(u2H + v 2H ), and 1 µs,t,u,v = s2H +t2H +u2H +v 2H − [(t+v )2H +|t−v |2H +(s+u)2H +|s−u|2H ], 2 for all s, t, u, v ≥ 0. In order to prove Theorem 5, we need the following propositions. Proposition 6. Under the assumptions above, the following statements are equivalent: 2 (i) H < d+2 ; TTTT d − 2 −1 2 (λs,t ρu,v − µ2 (ii) s,t,u,v ) µs,t,u,v dudv dsdt < ∞. 0000
12 Proof. By (3.12), we have λs,t ρu,v − µ2 4H f (x) + s4H f (y ) + t2H s2H g (x, y ) s,t,u,v = t t4H x2H (1 − x)2H + s4H y 2H (1 − y )2H + t2H s2H (x2H (1 − y )2H + y 2H (1 − x)2H ) = [x2H t2H + y 2H s2H ][(1 − x)2H t2H + (1 − y )2H s2H ]. (4.1) On the other hand, an elementary calculus can show that 1 1 x2H ≤ 1 + x2H − (1 + x)2H − (1 − x)2H ≤ (2 − 22H −1 )x2H 2 2 for all x, H ∈ (0, 1). By (3.5), we obtain (t2H x2H + s2H y 2H )2 ≤ µ2 2H −1 2 2H 2H ) (t x + s2H y 2H )2 . (4.2) s,t,u,v ≤ (2 − 2 It follows that T T T T − d −1 λs,t ρu,v − µ2 µ2 s,t,u,v dsdtdudv 2 s,t,u,v 0 0 0 0 T 1 T 1 (t2H x2H + s2H y 2H )st ≥ CH,T dy dsdxdt d ((1 − x)2H t2H + (1 − y )2H s2H )1+ 2 0 0 0 0 1 1 1 1 (t2H x2H + s2H y 2H )st ≥ CH,T dy dsdxdt d ((1 − x)2H t2H + (1 − y )2H s2H )1+ 2 0 0 0 0 y 1 x t s2H +1 x2H ≥ CH,T dy dx dt ds t2H (1+d/2)−1 (1 − x)2H (1+d/2) 0 0 0 0 y 1 1 x4−H (d−2) x4−H (d−2) (1 − x)1−2H (1+d/2) dx, ≥ CH,T dy dx = CH,T (1 − x)2H (1+d/2) 0 0 0 where CH,T > 0 is a constant depending only on H and T and its 2 value may diﬀer from line to line, which implies that H < d+2 if the convergence (ii) holds.
13 On the other hand, T T T T − d −1 λs,t ρu,v − µ2 µ2 s,t,u,v dudsdv dt 2 s,t,u,v 0 0 0 0 T 1 T 1 (t2H x2H + s2H y 2H )2 st ≤ CH dy dsdxdt [(x2H t2H + y 2H s2H )((1 − x)2H t2H + (1 − y )2H s2H )]d/2+1 0 0 0 0 T 1 T 1 (t2H x2H + s2H y 2H )2 st ≤ CH dy dsdxdt [(xH tH y H sH )((1 − x)H tH (1 − y )H sH )]d/2+1 0 0 0 0 T 1 T 1 T 4H ≤ CH dy dsdxdt d+2 d+2 d+2 d+2 H H H H t(d+2)H −1 s(d+2)H −1 x y (1 − x) (1 − y ) 2 2 2 2 0 0 0 0 0 is a constant depending only on H and its value may diﬀer from line to line. Thus, the proof is completed. Hence, Theorem 5 follows from the next proposition. Proposition 7. Under the assumptions above, the following statements are equivalent: T ∈ U if and only if T T T T d (λs,t ρu,v − µ2 − 2 −1 2 s,t,u,v ) µs,t,u,v dudv dsdt < ∞. (4.3) 0 0 0 0 In order to prove Proposition 7, we need some preliminaries(see Nu- alart [28]). Let X, Y be two random variables with joint Gaussian distribution such that E (X ) = E (Y ) = 0 and E (X 2 ) = E (Y 2 ) = 1. Then, for all n, m ≥ 0, we have 0, m = n, E (Hn (X )Hm (Y )) = (4.4) 1 n [E (XY )] , m = n. n! Moreover, elementary calculus can show that the following lemma holds. Lemma 8 ( [7]). Suppose d ≥ 1. For any x ∈ [−1, 1) we have ∞ n 2n(2k1 − 1)!! · · · · · (2kd − 1)!! n d x(1 − x)−( 2 +1) . x (2k1 )!! · · · · · (2kd )!! n=1 k1 ,...,kd =0 k1 +···+kd =n Particularly, this is an equality if and only if d = 1 (see An and Yan [32]).
14 It follows from µ2 s,t,u,v ≤ λs,t ρu,v that −( d +1) d µ2 µ2 µ2 1 2 2 s,t,u,v s,t,u,v s,t,u,v = 1− d λs,t ρu,v λs,t ρu,v λs,t ρu,v +1 (λs,t ρu,v − µ2 s,t,u,v ) 2 ∞ n µ2 n 2n(2k1 − 1)!! · · · · · (2kd − 1)!! s,t,u,v d. (2k1 )!! · · · · · (2kd )!! (λs,t ρu,v )n+ 2 n=1 k1 ,...,kd =0 k1 +···+kd =n Proof of Proposition 7. For ε > 0, T ≥ 0, we denote 2 ΦΘε (κ) := E (|Υ√κ ε,T | ) and ΦΘ (κ) := E (|Υ√κ T |2 ). Thus, by Proposition 2.1, it suﬃces to prove (4.3) if and only if ΦΘ (1) < ∞. Noticing that T T pε (StH − Ss )dsdt H = ε,T 0 0 T T 1 |ξ|2 H −S H ei ξ,St e−ε = dξ dsdt s 2 (2π )d 0 0 Rd T T ∞ ξ , StH − Ss H 1 1 2 e− 2 (λs,t +ε)|ξ| in σ n (t, s, ξ )Hn = dξ dsdt (2π )d σ (t, s, ξ ) n=0 0 0 Rd ∞ ≡ Fn . n=0 Thus, by (4.4) and Lemma 8, we have ∞ nE (|Fn |2 ) ΦΘε (1) = n=0  ∞ n  e− 2 ((λs,t +ε)|ξ| ) σ n (t, s, ξ )σ n (u, v, η ) 1 2 +(ρ 2 u,v +ε)|η | = E (2π )2d n=0 [0,T ]4 R2d ξ , StH − Ss H H H η , S u − Sv Hn Hn dξ dη dudv dsdt σ (t, s, ξ ) σ (u, v, η )
15 ∞ 1 e− 2 ((λs,t +ε)|ξ| ) ξ , η n dξ dη 1 2 +(ρ 2 u,v +ε)|η | µn = s,t,u,v dudv dsdt (2π )2d (n − 1)! n=1 [0,T ]4 R2d ∞ 1 e− 2 ((λs,t +ε)|ξ| ) ξ, η 1 2 +(ρ 2 u.v +ε)|η | µ2n dudv dsdt 2n = dξ dη s,t,u,v (2π )2d (2n − 1)! n=1 [0,T ]4 R2d ∞ 1 µ2n dudv dsdt = s,t,u,v (2π )2d (2n − 1)! n=1 [0,T ]4 e− 2 ((λs,t +ε)(ξ1 +···+ξd )+(ρu,v +ε)(η1 +···+ηd ) (ξ1 η1 + · · · + ξd ηd )2n dξ1 · · · dξd dη1 . . . dηd 1 2 2 2 2 × R2d ∞ 1 e− 2 ((λs,t +ε)(ξ1 +···+ξd )+(ρu,v +ε)(η1 +···+ηd )) 1 2 2 2 2 µ2n dudv dsdt × = s,t,u,v (2π )2d (2n − 1)! n=1 [0,T ]4 R2d n (ξ1 η1 )2k1 (ξ2 η2 )2k2 . . . (ξd ηd )2kd dξ1 . . . dξd dη1 . . . dηd k1 ,...,kd =0 k1 +···+kd =n ∞ n µ2n 1 2n(2k1 − 1)!! · · · · · (2kd − 1)!! s,t,u,v = dudv dsdt d (2π )d (2k1 )!! · · · · · (2kd )!! ((λs,t + ε)(ρu,v + ε))n+ 2 n=1 k1 ,...,kd =0 [0,T ]4 k1 +···+kd =n d µ2 2 − 2 −1 s,t,u,v ((λs,t + ε)(ρu,v + ε) − µs,t,u,v ) dudv dsdt, [0,T ]4 where we have used the following fact: ∞ 1 1 2k − 2 (λs,t +ε)ξ 2 2 ξ 2k e− 2 (λs,t +ε)ξ dξ ξe dξ = 2 0 R √ 1 k+ 1 1 1 (λs,t + ε)−(k+ 2 ) = 2π (2k − 1)!!(λs,t + ε)−(k+ 2 ) . =2 Γ k+ 2 2 It follows that d µ2 2 − 2 −1 lim ΦΘε (1) s,t,u,v (λs,t ρu,v − µs,t,u,v ) dudv dsdt ε→0 [0,T ]4 for all T ≥ 0. This completes the proof. 5. Regularity of the intersection local time The main object of this section is to prove the next theorem.
16 Theorem 9. Let Hd < 2. Then, the intersection local time admits t the following estimate: 2 ) ≤ Ct2−Hd |t − s|2−Hd , E (| t − s| for a constant C > 0 depending only on H and d. Proof. Let C > 0 be a constant depending only on H and d and its value may diﬀer from line to line. For any 0 ≤ r, l, u, v ≤ T , denote σ 2 = Var ξ Sr − SlH + η Su − Sv H H H . Then, the property of strong local nondeterminism (see Yan et al. [24]) : there exists a constant κ0 > 0 such that (see Berman [33]) the inequality n n StH StH−1 u2 Var StH − StH−1 . Var uj − ≥ κ0 (5.1) j j j j j j =2 j =2 holds for 0 ≤ t1 < t2 < · · · < tn ≤ T and uj ∈ R, j = 2, 3, . . . , n. and (1.2) yield σ 2 = Var ξ Sr − Su − ξ SlH − Sv + (ξ + η ) Su − Sv H H H H H ≥ C ξ 2 (|r − u|2H + |l − v |2H ) + (ξ + η )2 (u2H + v 2H ) . It follows from (3.1) that for 0 ≤ s ≤ t ≤ T t t t t 1 e− 2 (σ ) dξ dη 1 2 +ε|ξ |2 +ε|η |2 2 E | ε,t − ε,s | = drdl dudv (2π )2d s s s s R2d t t t s 4 e− 2 (σ ) dξ dη 1 2 +ε|ξ |2 +ε|η |2 + dr dl dudv (2π )2d s s s 0 R2d t s t s 4 e− 2 (σ ) dξ dη 1 2 +ε|ξ |2 +ε|η |2 + dr dl dudv (2π )2d s 0 s 0 R2d 1 ≡ [A1 (s, t) + 4A2 (s, t) + 4A3 (s, t)] . (2π )2d
17 We have t t t t e− 2 (σ ) dξ dη 1 2 +ε|ξ |2 +ε|η |2 A1 (s, t) = dr dl dudv s s s s R2d t t t t −d dudv (|r − u|2H + |l − v |2H )(u2H + v 2H ) ≤C drdl 2 s s s s t t t t Hd Hd Hd Hd |r − u|− |l − v |− u− v− ≤C drdldudv 2 2 2 2 s s s s  2  2 t t t r Hd Hd Hd Hd =C drdu ≤ 4C  dudr , |r − u|− u− (r − u)− u− 2 2 2 2 s s s s for 0 ≤ s ≤ t ≤ T . Noting that 1 (1 − m)x−1 mx−1 dm ≤ βx (1 − α)x , α for all α ∈ [0, 1] and x > 0, where βx is a constant depending only on x, we get t r t 1 − Hd − Hd Hd Hd r1−Hd dr (1 − m)− m− (r − u) u dudr = dm 2 2 2 2 s s s s/r 2−dH ≤ C (t − s) , which yields A1 (s, t) ≤ C (t − s)4−2dH ,
18 for 0 ≤ s ≤ t ≤ T . Similarly, for A2 (s, t) and A3 (s, t) we have also t t t s e− 2 (σ ) dξ dη 1 2 +ε|ξ |2 +ε|η |2 A2 (s, t) = dr dl dudv s s s 0 R2d t t t s −d dudv (|r − u|2H + |l − v |2H )(u2H + v 2H ) ≤C dr dl 2 s s s 0 t t t s − Hd − Hd Hd Hd |l − v |− v− =C |r − u| u drdu dl dv 2 2 2 2 ss s 0 2−Hd 2−Hd ≤ Ct (t − s) , t s t s e− 2 (σ ) dξ dη 1 2 +ε|ξ |2 +ε|η |2 A3 (s, t) = dr dl dudv s 0 s 0 R2d t s t s −d dudv (|r − u|2H + |l − v |2H )(u2H + v 2H ) ≤C drdl 2 s 0 s 0 t t s s − Hd − Hd Hd Hd |l − v |− v− =C |r − u| u drdu dldv 2 2 2 2 ss 0 0 2−Hd 2−Hd ≤ Ct (t − s) , for 0 ≤ s ≤ t ≤ T . Thus, Theorem 2 and Fatou’s lemma yield 2 2 2 ) ≤ Ct2−Hd (t − s)2−Hd . E (| t − s| ) = E (lim | − ε,s | ) ≤ lim inf E (| − ε,s | ε,t ε,t ε→0 ε→0 This completes the proof. Competing interests The author declare that he has no competing interests. Acknowledgements The author would like to thank anonymous earnest referee whose remarks and suggestions greatly improved the presentation of the pa- per. The author is very grateful to Professor Litan Yan for his valuable guidance. This work was supported by National Natural Science Foun- dation of China (Grant No. 11171062), Key Natural Science Foun- dation of Anhui Educational Committee (Grant No. KJ2011A139),
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