# Hölder Continuity and Occupation-Time Formulas for fBm Self-Intersection Local Time and Its Derivative

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## Abstract

We prove joint Hölder continuity and an occupation-time formula for the self-intersection local time of fractional Brownian motion. Motivated by an occupation-time formula, we also introduce a new version of the derivative of self-intersection local time for fractional Brownian motion and prove Hölder conditions for this process. This process is related to a different version of the derivative of self-intersection local time studied by the authors in a previous work.

## Keywords

Intersection local time Fractional Brownian motion Occupation-time formula## Mathematics Subject Classification (2010)

60G22 60J55## Notes

### Acknowledgments

We thank Yimin Xiao for helpful comments. The second author was supported by Australian Research Council Grant DP0988483.

## References

- 1.Chen X.: Random walk intersections: large deviations and related topics, vol 157. Amer Mathematical Society (2010)Google Scholar
- 2.Dynkin E.B.: Self-intersection gauge for random walks and for Brownian motion. Ann. Probab. 1–57 (1988)Google Scholar
- 3.Garsia A.M.: Topics in almost everywhere, convergence. Markham (1970)Google Scholar
- 4.Berman, S.: Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J.
**23**(1), 69–94 (1974)CrossRefGoogle Scholar - 5.Hu, Y.: Self-intersection local time of fractional Brownian motions–via chaos expansion. J. Math. Kyoto Univ.
**41**(2), 233–250 (2001)MATHMathSciNetGoogle Scholar - 6.Hu, Y., Nualart, D.: Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab.
**33**(3), 948–983 (2005)CrossRefMATHMathSciNetGoogle Scholar - 7.Hu, Y., Nualart, D.: Central limit theorem for the third moment in space of the Brownian local time increments. Electron. Commun. Probab.
**15**, 396–410 (2010)CrossRefMATHMathSciNetGoogle Scholar - 8.Jung P., Markowsky G.: On the tanaka formula for the derivative of self-intersection local time of fbm. Arxiv, preprint arXiv:1205.5551 (2012)Google Scholar
- 9.Markowsky G.: Proof of a Tanaka-like formula stated. In: Rosen J. (ed.) Séminaire XXXVIII. Séminaire de Probabilités XLI, pp. 199–202 (2008)Google Scholar
- 10.Markowsky, G.: Renormalization and convergence in law for the derivative of intersection local time in \(\mathbb{R}^2\). Stoch. Process. Appl.
**118**(9), 1552–1585 (2008)CrossRefMATHMathSciNetGoogle Scholar - 11.Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer Verlag, New York (1999)CrossRefMATHGoogle Scholar
- 12.Rogers, L.C.G., Walsh, J.B.: Local time and stochastic area integrals. Ann. Probab.
**19**(2), 457–482 (1991)CrossRefMATHMathSciNetGoogle Scholar - 13.Rosen, J.: The intersection local time of fractional Brownian motion in the plane. J. Multivar. Anal.
**23**(1), 37–46 (1987)CrossRefMATHGoogle Scholar - 14.Rosen, J.: Limit laws for the intersection local time of stable processes in \(R^2\). Stochastics
**23**(2), 219–240 (1988)CrossRefMATHMathSciNetGoogle Scholar - 15.Rosen J.: Derivatives of self-intersection local times. Séminaire de Probabilités XXXVIII, pp. 263–281 (2005)Google Scholar
- 16.Varadhan, S.R.S.: Appendix to Euclidian quantum field theory. In: Symanzy, K. (ed.) Local Quantum Theory. Academic Press, New York (1969)Google Scholar
- 17.Wu, D., Xiao, Y.: Regularity of intersection local times of fractional brownian motions. J. Theor. Probab.
**23**(4), 972–1001 (2010)CrossRefMATHMathSciNetGoogle Scholar - 18.Xiao, Y.: Holder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Probab. Theory Relat. Fields
**109**(1), 129–157 (1997)CrossRefMATHGoogle Scholar - 19.Yan, L., Yang, X., Lu, Y.: p-Variation of an integral functional driven by fractional Brownian motion. Stat. Probab. Lett.
**78**(9), 1148–1157 (2008)CrossRefMATHMathSciNetGoogle Scholar - 20.Yan, L., Liu, J., Yang, X.: Integration with respect to fractional local time with Hurst index \(1/2<H<1\). Potential Anal.
**30**(2), 115–138 (2009)CrossRefMATHMathSciNetGoogle Scholar

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