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

The Original Article was published on 23 November 2018

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

$$\begin{aligned} \delta_{1}&\geq k \bigl[(\bar{a}+\bar{b})\bar{c}+(\bar{b}+\bar{c}) \bar{a} \bigr] \\ &\geq k \bigl[ \bigl({a}^{2HK}+{b}^{2HK} \bigr){c}^{2HK}+ \bigl({b}^{2HK}+{c}^{2HK} \bigr){a}^{2HK} \bigr] \\ &\geq k \bigl[(a+b)^{2HK}{c}^{2HK}+(b+c)^{2HK}{a}^{2HK} \bigr], \end{aligned}$$

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

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Correspondence to Liheng Sang.

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The online version of the original article can be found under https://doi.org/10.1186/s13660-018-1916-3.

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Chen, Z., Sang, L. & Hao, X. Correction to: Renormalized self-intersection local time of bifractional Brownian motion. J Inequal Appl 2019, 21 (2019). https://doi.org/10.1186/s13660-019-1978-x

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