Correction to: Renormalized self-intersection local time of bifractional Brownian motion

  • Zhenlong Chen
  • Liheng SangEmail author
  • Xiaozhen Hao
Open Access

1 Correction

In the publication of this article [1], there are five errors. They have now been corrected in this correction.

The error:

1. Page 2, line -2–Page 3, line 1 : “The Dirac delta function is formally
$$\begin{aligned} \delta(x)=\lim_{\varepsilon\rightarrow 0}p_{\varepsilon}(x)=(2 \pi)^{-d} \int_{\mathbb{R}^{d}}\exp \bigl\{ i\langle \xi,x\rangle \bigr\} \,d{\xi}, \end{aligned}$$
Should instead read:
  • “In order to give a rigorous meaning to \(L(H,K,T)\), we approximate the Dirac delta function by the heat kernel”.

  • Remark: equation number “(1.6)” in line 3 of Page 3 and line 10 of Page 4 isn’t affected by the error.

The error:

2. Page 8, line 7: “\(\lambda=\lambda_{1}:=(a+b)^{2HK}, \rho=\rho_{1}:=(b+c)^{2HK}\)

Should instead read:

\(2^{-K}(a+b)^{2HK}\leq\lambda=\lambda_{1}\leq2^{1-K}(a+b)^{2HK}, 2^{-K}(b+c)^{2HK}\leq\rho=\rho_{1}\leq2^{1-K}(b+c)^{2HK} \).

The error:

3. Page 8, line 12: “\(\lambda=\lambda_{2} :=(a+b+c)^{2HK}, \rho=\rho_{2}:=b^{2HK}\),”

Should instead read:

\(2^{-K}(a+ b+c )^{2HK}\leq\lambda=\lambda_{2}\leq2^{1-K}(a+b+c)^{2HK}, 2^{-K}b^{2HK}\leq\rho=\rho_{2} \leq2^{1-K}b^{2HK}\).

The error:

4. Page 8, line 18: “\(\lambda=\lambda_{3} :=a^{2HK}, \rho=\rho_{3}:=c^{2HK}\)

Should instead read:

\(2^{-K} a^{2HK}\leq\lambda=\lambda_{3}\leq2^{1-K}a^{2HK}, 2^{-K}c^{2HK}\leq\rho=\rho_{3}\leq2^{1-K}c^{2HK}\),.

The error:

5. Page 10, Line -4–Page 11, line 6. Should instead read:

$$\begin{aligned} \lambda_{1} \bar{c}+\rho_{1} \bar{a}\geq\frac{1}{2}( \bar{a}\bar{b}+\bar {b}\bar{c}+\bar{a}\bar{c}), \end{aligned}$$
when k is small enough, we have
$$\begin{aligned} \delta_{1}&\geq k \bigl[(\bar{a}+\bar{b})\bar{c}+(\bar{b}+\bar{c}) \bar{a} \bigr] \\ &\geq k \bigl[ \bigl({a}^{2HK}+{b}^{2HK} \bigr){c}^{2HK}+ \bigl({b}^{2HK}+{c}^{2HK} \bigr){a}^{2HK} \bigr] \\ &\geq k \bigl[(a+b)^{2HK}{c}^{2HK}+(b+c)^{2HK}{a}^{2HK} \bigr], \end{aligned}$$



  1. 1.
    Chen, Z., Sang, L., Hao, X.: Renormalized self-intersection local time of bifractional Brownian motion. J. Inequal. Appl. 2018, 326 (2018). MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.School of Statistics and MathematicsZhejiang Gongshang UniversityHangzhouChina
  2. 2.School of Mathematics and FinanceChuzhou UniversityChuzhouChina

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