Renormalized self-intersection local time of bifractional Brownian motion

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B^{H,K}=\{B^{H,K}(t), t \geq 0\}$\end{document}BH,K={BH,K(t),t≥0} be a d-dimensional bifractional Brownian motion with Hurst parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H\in (0,1)$\end{document}H∈(0,1) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K\in (0,1]$\end{document}K∈(0,1]. Assuming \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d\geq 2$\end{document}d≥2, we prove that the renormalized self-intersection local time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \int^{T}_{0} \int^{t}_{0}\delta \bigl(B^{H,K}(t)-B^{H,K}(s) \bigr)\,ds\,dt-\mathbb{E} \biggl( \int^{T}_{0} \int^{t}_{0}\delta \bigl(B^{H,K}(t)-B^{H,K}(s) \bigr)\,ds\,dt \biggr) \end{aligned}$$ \end{document}∫0T∫0tδ(BH,K(t)−BH,K(s))dsdt−E(∫0T∫0tδ(BH,K(t)−BH,K(s))dsdt) exists in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}$\end{document}L2 if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$HKd< 3/2$\end{document}HKd<3/2, where δ denotes the Dirac delta function. Our work generalizes the result of the renormalized self-intersection local time for fractional Brownian motion.


Introduction
Fractional Brownian motion has received much attention in recent years due to its longrange dependence, stationarity increments, and self-similarity. It has been widely applied in turbulence, image processing, mathematics finance, and so on for small increments. However, it is inadequate to large increments. So, it is very natural to explore the extension of fractional Brownian motion which keeps some properties of fractional Brownian motion (gaussianity, stationarity of small increments, self-similarity), and then bifractional Brownian motion as a generalization of fractional Brownian motion has been investigated by many authors, see [5,13,15] and the references therein for more details.
Let us briefly recall some related definitions of bifractional Brownian motion as follows. Set B H,K 0 = {B H,K 0 (t), t ≥ 0} be a bifractional Brownian motion in R with Hurst parameters H ∈ (0, 1) and K ∈ (0, 1], i.e., a centered, real-valued Gaussian process with zero mean and covariance function given by This process is HK -self similar and satisfies the following estimates: for each T > 0 and s, t ∈ [0, T]. Moreover, we can easily check that it is Hölder continuous of order δ for any δ < HK from the Kolmogorov criterium. In particular, if K = 1, B H,1 0 (t) is a fractional Brownian motion with Hurst parameter H ∈ (0, 1).
We associate with B H,K 0 a Gaussian process B H,K = {B H,K (t), t ≥ 0} in R d by On the other hand, since the work of Varadhan [16], self-intersection local time, as an important topic of probability theory, has been widely considered and studied in recent years. Especially, when it comes to Brownian motion and fractional Brownian motion, it has been extensively studied, see [1,2,4,6,10,11,17] and the references therein.
Recently, the self-intersection local time of bifractional Brownian motion has already been researched by few scholars. Jiang and Wang [9] studied the existence and smoothness of the self-intersection local time of bifractional Brownian motions. Chen et al. [3] considered the existence and smoothness of self-intersection local times for a large class of Gaussian random fields, including fractional Brownian motion, fractional Brownian sheets, and bifractional Brownian motion. For more on the local time of bifractional Brownian motion, we can see [14,18] and so on.
We know that the non-renormalized self-intersection local time of fractional Brownian motion exists in L 2 for Hd < 1 by the results of Jiang and Wang [9] and Chen et al. [3]. But for the case of renormalization, Hu and Nualart [7] obtained that the renormalized self-intersection local time of fractional Brownian motion exists in L 2 for Hd < 3/2. Therefore, the existence is different between renormalization and non-renormalization of self-intersection local time. In this paper, we consider the existence of renormalized self-intersection local time for bifractional Brownian motion. Our conclusions generalize the result of fractional Brownian motion in Hu and Nualart [7] to bifractional Brownian motions.
In this paper, the following local times of bifractional Brownian motion will be involved, including the local time H,K T (x) and the self-intersection local time I(H, K, T) of bifractional Brownian motion B H,K (t). Formally, they are defined respectively as follows: for T > 0, (1.4) and where δ(x) is the Dirac delta function for x ∈ R d . The Dirac delta function is formally By (1.6), we define the approximated self-intersection local time of bifractional Brownian motion by (1.8) We will consider the following two questions: (1) We consider the existence in L 2 and a sharp upper bound of second moment of local time H,K T (x) for bifractional Brownian motion. Although the existence of the local time for anisotropic Gaussian random fields is obtained in Theorem 2.6 by Chen et al. [3], which contains the result of bifractional Brownian motion, a sharp upper bound of second moment of the local time for anisotropic Gaussian random fields is not got. It is not enough to research the local time for Gaussian random fields. So, in order to fill this vacancy for bifractional Brownian motion, we give the following Theorem 1.
where k is a constant depending on H and K .
(2) The latter problem is to generalize the result of Hu and Nualart [7] to bifractional Brownian motion. That is, we will consider the existence of the renormalized self-intersection local time of bifractional Brownian motion in L 2 . We get the following Theorem 1.2. The paper is organized as follows. In Sect. 2, we study the square-integrable of the local time of d-dimensional bifractional Brownian motion. We prove the existence of the selfintersection local time of d-dimensional bifractional Brownian motion in Sect. 3.
For simplicity, we will use k to denote unspecified positive finite constants which may be different in each appearance throughout this paper.

Square integrable of the local time
In this section, the local time of the d-dimension bifractional Brownian motion will be discussed. We firstly give the following lemma which plays an important role in proving the existence of the local time and Theorem 1.1. ([15], Proposition 2.1) For all constants 0 < a < b, B H,K 0 (t) is strongly locally ϕ-nondeterministic on I = [a, b] with ϕ(r) = r 2HK . That is, there exist positive constants k and r 0 such that, for all t ∈ I and all 0 < r ≤ min{t, r 0 },
where we used the fact that E[e iX ] = exp{-1 2 Var(X)} for any Gaussian random variable X. By the local nondeterminism (2.2) of bifractional Brownian motion and Var(B H,K (t)) = t 2HK , we have that, for 0 ≤ s < t ≤ T, there is a positive constant k > 0 such that Therefore, we get that the last integral of (2.3) is bounded by the following expression: By integrating with respect to ξ and η, respectively, and changing the variable s = tu for s, we obtain that expression (2.4) is equal to This completes the proof.
Remark this proposition implies that the local time of bifractional Brownian motion exists in L 2 if HKd < 1. This is consistent with Theorem 2.6 in [3] and Theorem 1 in [12].

The existence of the renormalized self-intersection local time
In this section, we will prove the existence of the renormalized self-intersection local time of bifractional Brownian motion, which extends the result of Hu and Nualart [7] to bifractional Brownian motion. For more on the existence of the self-intersection local time of bifractional Brownian motion, we can refer to Jiang and Wang [9] and Chen et al. [3]. According to the definition for the self-intersection local time of bifractional Brownian motion and (1.7), we get Then, by the independence of B H,K 1 , . . . , B H,K d , we obtain the mean of the self-intersection local time (3.2) and the second moment of the self-intersection local time where B H,K 1 (t) denotes a one-dimensional bifractional Brownian motion with Hurst parameters H and K , λ is the variance of B H,K Therefore, the limit of as ε 1 , ε 2 tend to 0, exists if and only if T [(λρμ 2 ) -d 2 -(λρ) -d 2 ] dτ . By Loève's criterion of mean-square convergence, we know that a necessary and sufficient condition for the convergence of L ε (H, K, T) -E[L ε (H, K, T)] in L 2 is the existence of finite limit of (3.5) as ε 1 , ε 2 tend to 0. Consequently, a necessary and sufficient condition for the convergence of L ε (H, For notation and simplicity along the paper, we will denote δ = λρμ 2 and Θ = δ -d 2 -(λρ) -d 2 , then the last inequality is rewritten as For simplicity, we give some symbols. The region T = {(s, t, s , t ) | 0 ≤ s < t ≤ T, 0 ≤ s < t ≤ T} is decomposed as follows: Then, for (s, t, s , t ) ∈ T , we can consider the following three cases: (iii) If (s, t, s , t ) ∈ T 3 , denoting a = ts, b = st, c = ts , and e = s, we have For the term 3,1 , we easily get 3,1 ≤ kacb 2HK-2 . Now, we consider the term 3,2 . Let So, by the mean theorem, there exist ξ ∈ (e, e + a) and η ∈ (e + a + b, e + a + b + c) such that where we used the fact that ξ 2H + η 2H ≥ ξ 2Hα η 2Hη , α = 1-2H 2H(K-2) , β = 1 -1-2H 2H(K-2) . Further, Combining the upper bounds of 3,1 and 3,2 , we get that μ 3 ≤ kb 2HK-2 ac.
The following lemma provides some useful inequalities for the proof of Theorem 1.2.

Lemma 3.2 Following the decomposition of the region T , there exists a constant k such that
On the other hand, by using inequality (1.2), we get Thus, through combining (3.12) with (3.13), it is easy to get This implies that (λ 1kākb)u 2 + 2(μ 1kb)uv + (ρ 1kbkc)v 2 ≥ 0.
By Lemma 3.2 and δ i , Θ i , i = 1, 2, 3, defined above, we can get the following result.

Lemma 3.3 For i = 2, 3, there exists a constant k such that
and The proof of this lemma can be found in Hu and Nualart [7]. For proving Theorem 1.2, we will make use of the following elementary lemma. (3.7). Then Ξ T < +∞ if and only if HKd < 3 2 .

Lemma 3.4 Let Ξ T be defined by
Proof The proof will be done in two steps.
Step 1. We give the proof of the sufficient condition, that is, if HKd < 3 2 , we claim that We split the proof into three cases for the value of i.
If HK d ≥ 1, the last integral of (3.25) is written by For i = 3, we also decompose the integral region as T 3 = I 1 + I 2 + I 3 + I 4 , where for some fixed but arbitrary η 1 > 0 and η 2 > 0.
Firstly, we consider in the region I 1 . By (3.19) of Lemma 3.3, it follows that Secondly, in the region I 2 . By (3.8) and (3.18), we obtain that where we have used the inequality -2 3 dHK < 2 -2HK -dHK . Therefore, Finally, we consider the case a ≥ η 2 b and c < η 1 b, the region I 4 can be achieved similarly.
It follows that