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  • Correction
  • Open Access

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

Journal of Inequalities and Applications20192019:21

https://doi.org/10.1186/s13660-019-1978-x

  • Received: 19 December 2018
  • Accepted: 19 December 2018
  • Published:

The original article was published in Journal of Inequalities and Applications 2018 2018:326

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
$$\begin{aligned} \delta(x)=\lim_{\varepsilon\rightarrow 0}p_{\varepsilon}(x)=(2 \pi)^{-d} \int_{\mathbb{R}^{d}}\exp \bigl\{ i\langle \xi,x\rangle \bigr\} \,d{\xi}, \end{aligned}$$
(1.6)
where”
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:

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

Notes

Declarations

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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’ Affiliations

(1)
School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, China
(2)
School of Mathematics and Finance, Chuzhou University, Chuzhou, China

References

  1. Chen, Z., Sang, L., Hao, X.: Renormalized self-intersection local time of bifractional Brownian motion. J. Inequal. Appl. 2018, 326 (2018). https://doi.org/10.1186/s13660-018-1916-3 MathSciNetView ArticleGoogle Scholar

Copyright

© The Author(s) 2019

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