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/2011/1/139 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 Journal of Inequalities and Applications © 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  T = T  0 T  0 δ  S H t −  S H s  dsdt, T > 0, where δ denotes the Dirac delta function at zero. By elementary inequalities, we show that  T exists in L 2 if and only if Hd < 2 and it is smooth in the sense of the Meyer-Watanabe if and only if H < 2 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 = {S H t , t ≥ 0} with S H 0 = 0 and C H (s, t) := E  S H t S H s  = s 2H + t 2H − 1 2  (s + t) 2H + |t −s| 2H  (1.1) for all s, t ≥ 0. For H = 1 2 , S H 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 2H−1 )∧1](t−s) 2H ≤ E   S H t − S H s  2  ≤ [(2−2 2H−1 )∨1](t−s) 2H . (1.2) Thus, Kolmogorov’s continuity criterion implies that sub-fBm is H¨older 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 R d , d ≥ 2, with the same indices H ∈ (0, 1). This means that we have two d-dimensional independent centered Gauss- ian processes S H = {S H t , t ≥ 0} and  S H = {  S H t , t ≥ 0} with covariance structure given by E  S H,i t S H,j s  = E   S H,i t  S H,j s  = δ i,j C H (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  0 T  0 δ  S H t −  S H s  dsdt, (1.3) 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 p ε (x) = (2πε) − d 2 e − |x| 2 2ε , x ∈ R d . Then, we can consider the following family of random variables indexed by ε > 0  ε,T = T  0 T  0 p ε (S H t −  S H s )dsdt, (1.4) 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 2 , the process S H and  S H are Brownian motions. The inter- section local time of independent Brownian motions has been studied by several authors (see Wolpert [2], Geman et al. [3] and the references therein). In the general case, that is H = 1 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 2 and 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 H < 2 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 L 2 (Ω, P). We refer to Meyer [27] and Nualart [28] and Hu [29] and the references therein for more details. Let X = {X t , t ∈ [0, T ]} be a d−dimensional Gaussian process defined on the probability space (Ω, F, P ) with mean zero. If p n (x 1 , . . . , x k ) is a polynomial of degree n of k variables x 1 , . . . , x k , then we call p n (X i 1 t 1 , . . . , X i k t k ) a polynomial functional of X with t 1 , . . . , t k ∈ [0, T ] and 1 ≤ i 1 , . . . , i k ≤ d. Let P n be the completion with respect to the 4 L 2 (Ω, P) norm of the set {p m (X i 1 t 1 , . . . , X i k t k ) : 0 ≤ m ≤ n}. Clearly, P n is a subspace of L 2 (Ω, P). If C n denotes the orthogonal complement of P n−1 in P n , then L 2 (Ω, P) is actually the direct sum of C n , i.e., L 2 (Ω, P) = ∞  n=0 C n . (2.1) For F ∈ L 2 (Ω, P), we then see that there exists F n ∈ C n , n = 0, 1, 2, . . . , such that F = ∞  n=0 F n , (2.2) This decomposition is called the chaos expansion of F . F n is called the n-th chaos of F . Clearly, we have E(|F | 2 ) = ∞  n=0 E(|F n | 2 ). (2.3) As in the Malliavin calculus, we introduce the space of “smooth” functionals in the sense of Meyer and Watanabe (see Watanabe [30]): U := {F ∈ L 2 (Ω, P) : F = ∞  n=0 F n and ∞  n=0 nE(|F n | 2 ) < ∞}, and F ∈ L 2 (Ω, P) is said to be smooth if F ∈ U . Now, for F ∈ L 2 (Ω, P), we define an operator Υ u with u ∈ [0, 1] by Υ u F := ∞  n=0 u n F n . (2.4) Set Θ(u) := Υ √ u F . Then, Θ(1) = F. Define Φ Θ (u) := d du (||Θ(u)|| 2 ), where ||F || 2 := E(|F | 2 ) for F ∈ L 2 (Ω, P). We have Φ Θ (u) = ∞  n=1 nu n−1 E(|F n | 2 ). (2.5) Note that ||Θ(u)|| 2 = E(|Θ(u)| 2 ) =  ∞ n=1 E(u n |F n | 2 ). Proposition 1. Let F ∈ L 2 (Ω, P). Then, F ∈ U if and only if Φ Θ (1) < ∞. Now consider two d-dimensional independent sub-fBms S H and  S H with indices H ∈ (0, 1). Let H n (x), x ∈ R be the Hermite polynomials of degree n. That is, H n (x) = (−1) n 1 n! e x 2 2 ∂ n ∂x n e − x 2 2 . (2.6) 5 Then, e tx− t 2 2 = ∞  n=0 t n H n (x) (2.7) for all t ∈ C and x ∈ R, which deduces exp(iuξ, S H t −  S H s  + 1 2 u 2 |ξ| 2 Var(S H,1 t −  S H,1 s )) = ∞  n=0 (iu) n σ n (t, s, ξ)H n  ξ, S H t −  S H s  σ(t, s, ξ)  , where σ (t, s, ξ) =  Var(S H,1 t −  S H,1 s )|ξ| 2 for ξ ∈ R d . Because of the orthogonality of {H n (x), x ∈ R} n∈Z + , we will get from (2.2) that (iu) n σ n (t, s, ξ)H n  ξ, S H t −  S H s  σ(t, s, ξ)  is the n-th chaos of exp  iuξ, S H t −  S H s  + 1 2 u 2 |ξ| 2 Var  S H,1 t −  S H,1 s   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 2 and d ≥ 2. We have obtained the following result. Theorem 2. (i) If Hd < 2, then the  ε,T converges in L 2 (Ω). The limit is denoted by  T (ii) If Hd ≥ 2, then lim ε→0 E( ε,T ) = +∞, and lim ε→0 Var( ε,T ) = +∞. Note that if {S 1 2 t } t≥0 is a planar Brownian motion, then  ε = T  0 T  0 p ε  S 1/2 t − S 1/2 s  dsdt, diverges almost sure, when ε tends to zero. Varadhan, in [31], proved that the renormalized self-intersection local time defined as lim ε→0 ( ε − 6 E ε ) exists in L 2 (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 2 (Ω), and therefore,  T , the intersection local time of S H and  S H , does not exist. Using the following classical equality p ε (x) = 1 (2πε) d 2 e − |x| 2 2ε = 1 (2π) d  R d e iξ,x e −ε |ξ| 2 2 dξ, we have  ε,T = T  0 T  0 p  (S H t −  S H s )dsdt = 1 (2π) d T  0 T  0  R d e iξ,S H t −  S H s  · e −ε |ξ| 2 2 dξdsdt. (3.1) Since ξ, S H t −  S H s  ∼ N(0, |ξ| 2 (2 − 2 2H−1 )(t 2H + s 2H )), so E[e iξ,S H t −  S H s  ] = e −[(2−2 2H−1 )(t 2H +s 2H )] |ξ| 2 2 . Therefore, E( ε,T ) = 1 (2π) d T  0 T  0  R d E[e iξ,S H t −  S H s  ] · e −ε |ξ| 2 2 dξdsdt = 1 (2π) d T  0 T  0  R d e −[ε+(2−2 2H−1 )(t 2H +s 2H )] |ξ| 2 2 dξdsdt = 1 (2π) d 2 T  0 T  0 [ε + (2 −2 2H−1 )(t 2H + s 2H )] − d 2 dsdt, (3.2) where we have used the fact  R d e −[ε+(2−2 2H−1 )(t 2H +s 2H )] |ξ| 2 2 dξ =  2π ε + (2 −2 2H−1 )(t 2H + s 2H )  d 2 . 7 We also have E( 2 ε,T ) = 1 (2π) 2d  [0,T ] 4  R 2d E  e i  ξ,S H t −  S H s  +i  η,S H u −  S H v   × e − ε(|ξ| 2 +|η| 2 ) 2 dξdηdsdtdudv. (3.3) Let we intro duce some notations that will be used throughout this paper, λ s,t = Var(S H,1 t − S H,2 s ) = (2 − 2 2H−1 )(t 2H + s 2H ), ρ u,v = Var(S H,1 v − S H,2 u ) = (2 − 2 2H−1 )(u 2H + v 2H ), and µ s,t,u,v = Cov  S H,1 t − S H,2 s , S H,1 v − S H,2 u  = s 2 H + t 2 H + u 2 H + v 2 H − 1 2 [(t + v) 2 H + |t −v| 2 H + (s + u) 2 H + |s −u| 2 H ], 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 E( 2 ε,T ) = 1 (2π) 2d  [0,T ] 4  R 2d exp  − 1 2  (λ s,t + ε)|ξ| 2 + (ρ u,v + ε)|η| 2 + 2µ s,t,u,v ξ, η   × dξdηdsdtdudv = 1 (2π) d  [0,T ] 4  (λ s,t + ε)(ρ u,v + ε) −µ 2 s,t,u,v  − d 2 dsdtdudv. (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 ρ u,v and µ s,t,u,v as following. ρ u,v = (2 − 2 2H−1 )(x 2H t 2H + y 2H s 2H ), µ s,t,u,v = t 2H  1 + x 2H − 1 2 [(1 + x) 2H + (1 −x) 2H ]  + s 2H  1 + y 2H − 1 2 [(1 + y) 2H + (1 −y) 2H ]  . (3.5) It follows that λ s,t ρ u,v − µ 2 s,t,u,v = t 4H f(x) + s 4H f(y) + t 2H s 2H g(x, y), (3.6) 8 where f(x) := (2 − 2 2H−1 ) 2 x 2H −  1 + x 2H − 1 2 (1 + x) 2H − 1 2 (1 − x) 2H  2 , and g(x, y) =(2 − 2 2H−1 ) 2  x 2H + y 2H  − 2  1 + x 2H − 1 2 (1 + x) 2H − 1 2 (1 − x) 2H  ×  1 + y 2H − 1 2 (1 + y) 2H − 1 2 (1 − y) 2H  . (3.7) For simplicity throughout this paper, we assume that the notation F  G means that there are positive constants c 1 and c 2 so that c 1 G(x) ≤ F (x) ≤ c 2 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)  x 2H (1 − x) 2H , (3.8) and g(x, y)  x 2H (1 − y) 2H + y 2H (1 − x) 2H (3.9) for all x, y ∈ [0, 1]. Lemma 4. Let A T :=  [0,T ] 4  λ s,t ρ u,v − µ 2 s,t,u,v  − d 2 dsdtdudv. Then, A T 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 D ε ⊂ [0, T] 4 , where D ε ≡  (s, t, u, v) ∈ R 4 + : s 2 + t 2 + u 2 + v 2 ≤ ε 2  . We make a change to spherical coordinates as following          s = r cos ϕ 1 , t = r sin ϕ 1 cos ϕ 2 , u = r sin ϕ 1 sin ϕ 2 cos ϕ 3 , v = r sin ϕ 1 sin ϕ 2 sin ϕ 3 . (3.10) 9 where 0 ≤ r ≤ ε, 0 ≤ ϕ 1 , ϕ 2 ≤ π, 0 ≤ ϕ 3 ≤ 2π, J = ∂(s, t, u, v) ∂(r, ϕ 1 , ϕ 2 , ϕ 3 ) = r 3 sin 2 ϕ 1 sin ϕ 2 . As λ s,t ρ u,v −µ 2 s,t,u,v is always positive, and λ s,t ρ u,v −µ 2 s,t,u,v = r 4H φ(θ), we have A T ≥  D ε (λ s,t ρ u,v − µ 2 s,t,u,v ) − d 2 dsdtdudv = ε  0 r 3−2Hd  Θ φ(θ)dθ, (3.11) 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 A T = +∞. Now, we turn to the proof of sufficient condition. Suppose that Hd < 2. By symmetry, we have A T = 4  Υ (λ s,t ρ u,v − µ 2 s,t,u,v ) − d 2 dsdtdudv, where Υ = {(u, v, s, t) : 0 < u < s ≤ T, 0 < v < t ≤ T }. By Lemma 3, we get λ s,t ρ u,v − µ 2 s,t,u,v = t 4H f(x) + s 4H f(y) + t 2H s 2H g(x, y)  t 4H x 2H (1 − x) 2H + s 4H y 2H (1 − y) 2H + t 2H s 2H (x 2H (1 − y) 2H + y 2H (1 − x) 2H ) = [x 2H t 2H + y 2H s 2H ][(1 − x) 2H t 2H + (1 −y) 2H s 2H ] = (v 2H + u 2H )[(t − v) 2H + (s −u) 2H ]. (3.12) These deduce for all H ∈ (0, 1) and T > 0, Λ T ≤ C H T  0 dt t  0 (v H (t − v) H ) −d/2 dv T  0 ds s  0 (u H (s − u) H ) −d/2 du = C H   T  0 t 1−Hd dt 1  0 x − Hd 2 (1 − x) − Hd 2 dx   2 < ∞.  Proof of Theorem 2. Suppose Hd < 2, we have E( ε,T ·  η,T ) = 1 (2π) d  [0,T ] 4 ((λ s,t + ε)(ρ u,v + η) − µ 2 s,t,u,v ) − d 2 dsdtdudv. [...]... 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→0 ε,T ) = +∞ 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. .. Ortiz-Latorre, S: Intersection local time for two independent fractional Brownian motions J Theor Probab 20:759–767 (2007) [9] Rosen, J: The intersection local time of fractional Brownian motion in the plane J Multivar Anal 23:7–46 (1987) [10] Wu, D, Xiao, Y: Regularity of intersection local times of fractional Brownian motions J Theor Probab 23:972–1001 (2010) [11] Symanzik, K: Euclidean quantum field theory... 62:487–550 (1940) e [2] Wolpert, R: Wiener path intersections and local time J Funct Anal 30:329– 340 (1978) [3] Geman, D, Horowitz, J, Rosen, J: A local time analysis of intersections of Brownian paths in the plane Ann Probab 12:86–107 (1984) [4] Imkeller, P, P´rez–Abreu, V, Vives, J: Chaos expansion of double intersection e local time of Brownian motion in Rd and renormalization Stoch Process Appl 56:1–34... dudvdsdt 16 Theorem 9 Let Hd < 2 Then, the intersection local time the following estimate: E(| t − s| 2 t admits ) ≤ Ct2−Hd |t − s|2−Hd , 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 H H H σ 2 = Var ξ Sr − SlH + η Su − Sv Then, the property of strong local nondeterminism... 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... Jost, R (ed.) Local Quantum Theory Academic Press, New York (1969) [12] Wolpert, R: Local time and a particle picture for Euclidean field theory J Funct Anal 30:341–357 (1978) [13] Stoll, A: Invariance principle for Brownian local time and polymer measures Math Scand 64:133–160 (1989) [14] Albeverio, S, Fenstad, JE, Høegh-Krohn, R, Lindstrøm, T: Nonstandard Methods in Stochastic Analysis and Mathematical... T, Streit, L, Watanabe, H: Intersection local times as generalized white noise functionals Acta Appl Math 46:351–362 (1997) ˜ [6] Albeverio, S, JoAo Oliveira, M, Streit, L: Intersection local times of independent Brownian motions as generalized White noise functionals Acta Appl Math 69:221–241 (2001) [7] Chen, C, Yan, L: Remarks on the intersection local time of fractional Brownian motions Stat Probab... used the following fact: ∞ 1 2k − 2 (λs,t +ε)ξ 2 ξ e 2 0 R =2 1 ξ 2k e− 2 (λs,t +ε)ξ dξ dξ = 2 k+ 1 2 1 Γ k+ 2 1 (λs,t + ε)−(k+ 2 ) = √ 1 2π(2k − 1)!!(λs,t + ε)−(k+ 2 ) It follows that d 2 − 2 −1 dudvdsdt µ2 s,t,u,v (λs,t ρu,v − µs,t,u,v ) lim ΦΘε (1) 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... Euclidean quantum field theory, by K Symanzik In: Jost, R (ed.) Local Quantum Theory Academic Press, New York (1969) [32] An, L, Yan, L: Smoothness for the collision local time of fractional Brownian motion Preprint (2009) [33] Berman, SM: Local nondeterminism and local times of Gaussian processes Indiana Univ Math J 23:69–94 (1973) ... by sub-fractional Brownian motion J Korean Stat Soc 40:337–346 (2011) [22] Shen, G, Chen, C: Stochastic integration with respect to the sub-fractional Brownian motion with H ∈ (0, 1 ) Stat Probab Lett 82:240–251(2012) 2 [23] Tudor, C: Some properties of the sub-fractional Brownian motion Stochastics 79:431–448 (2007) [24] Yan, L, Shen, G: On the collision local time of sub-fractional Brownian Motions . 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. + 1 2 u 2 |ξ| 2 Var  S H,1 t −  S H,1 s   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 2 and d ≥. µ 2 s,t,u,v ) − d 2 −1 dudvdsdt 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