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

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.

This is a preview of subscription content, access via your institution.

Fig. 1

References

  1. 1.

    Chen X.: Random walk intersections: large deviations and related topics, vol 157. Amer Mathematical Society (2010)

  2. 2.

    Dynkin E.B.: Self-intersection gauge for random walks and for Brownian motion. Ann. Probab. 1–57 (1988)

  3. 3.

    Garsia A.M.: Topics in almost everywhere, convergence. Markham (1970)

  4. 4.

    Berman, S.: Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23(1), 69–94 (1974)

    Article  Google Scholar 

  5. 5.

    Hu, Y.: Self-intersection local time of fractional Brownian motions–via chaos expansion. J. Math. Kyoto Univ. 41(2), 233–250 (2001)

    MATH  MathSciNet  Google Scholar 

  6. 6.

    Hu, Y., Nualart, D.: Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab. 33(3), 948–983 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. 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)

    Article  MATH  MathSciNet  Google Scholar 

  8. 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)

  9. 9.

    Markowsky G.: Proof of a Tanaka-like formula stated. In: Rosen J. (ed.) Séminaire XXXVIII. Séminaire de Probabilités XLI, pp. 199–202 (2008)

  10. 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)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer Verlag, New York (1999)

    Google Scholar 

  12. 12.

    Rogers, L.C.G., Walsh, J.B.: Local time and stochastic area integrals. Ann. Probab. 19(2), 457–482 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Rosen, J.: The intersection local time of fractional Brownian motion in the plane. J. Multivar. Anal. 23(1), 37–46 (1987)

    Article  MATH  Google Scholar 

  14. 14.

    Rosen, J.: Limit laws for the intersection local time of stable processes in \(R^2\). Stochastics 23(2), 219–240 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  15. 15.

    Rosen J.: Derivatives of self-intersection local times. Séminaire de Probabilités XXXVIII, pp. 263–281 (2005)

  16. 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. 17.

    Wu, D., Xiao, Y.: Regularity of intersection local times of fractional brownian motions. J. Theor. Probab. 23(4), 972–1001 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. 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)

    Article  MATH  Google Scholar 

  19. 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)

    Article  MATH  MathSciNet  Google Scholar 

  20. 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)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

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

Author information

Affiliations

Authors

Corresponding author

Correspondence to Greg Markowsky.

Appendix: Continuity for \(A\subset \mathcal{D}\) when \(H>1/2\)

Appendix: Continuity for \(A\subset \mathcal{D}\) when \(H>1/2\)

In the introduction we conjectured that

$$\begin{aligned} \hat{\alpha }_{t}^{\prime }(y) - t \mathbf{E}[\hat{\alpha }_{t}^{\prime }(y)] \end{aligned}$$
(3.1)

has a spatially continuous modification for \(1/2<H<2/3\). Here, we very briefly explain the reasoning behind this along with two possible avenues toward a proof (for a more thorough explanation, see the version of this paper on arxiv.org). Rosen [15] proved this in the case \(H=1/2\) by considering a generalization of (1.6) to subsets \(A\subset \mathcal{D}\). More specifically, define

$$\begin{aligned} \hat{\alpha }_{t}^{\prime }(y,A):= -\int \!\int \limits _A \delta ^{\prime }\left(B^H_s-B^H_r-y\right) \,dr \,ds. \end{aligned}$$
(3.2)

For

$$\begin{aligned} A_k^j := [(2k-2)2^{-j},(2k-1)2^{-j}] \times [(2k-1)2^{-j},(2k)2^{-j}], \end{aligned}$$
(3.3)

one can show that \(\hat{\alpha }_{t}^{\prime }(y,A^j_k)\) exists and is jointly continuous in \(y\) and \(t\). Note that \(\mathcal{D}= \cup _{j=1}^{\infty } \cup _{k=1}^{2^{j-1}} A_k^j\), and observe that when \(H=1/2\) and \(j\) is fixed,

$$\begin{aligned} \left\{ \alpha _{t,\varepsilon }\left(y,A_{k}^{j}\right)\right\} _{1\le k\le 2^{j-1}} \end{aligned}$$
(3.4)

are independent. In [15], this independence was used together with the following lemma [3, Proposition 3.5.2], to establish \(L^p\) bounds and Hölder continuity for \(\alpha _{t,\varepsilon }(y, A)\) which sufficed to show Kolmogorov’s continuity criterion for (3.1).

Lemma 1

Suppose \(X_1, \ldots , X_n\) are independent with \(E[X_j]=0\) for all \(j\) and \(M = \max _{1 \le j \le n} \mathbf{E}[X_j^{2p}] < \infty \), with \(p\) a positive integer. Let \(a_1, \ldots , a_n \in \mathbb{R }\). Then

$$\begin{aligned} E\left[|a_1 X_1 + \ldots + a_n X_n|^{2p}\right] \le C(p)M\left(a_1^2 + \ldots + a_n^2\right)^p. \end{aligned}$$
(3.5)

The difficulty when \(1/2<H<2/3\) is that we no longer have independence in (3.4); however, it may be that the local nondeterminism of fBm is enough. Perhaps a substitute for the above lemma can be deduced under the weaker condition of local nondeterminism, and this could be used to prove the conjecture.

In order to get an \(L^p\) bound for a single set \(A_k^j\) one can, for example, use the arguments in the proof of Theorem 1.2 and the Cauchy–Schwarz inequality to show the following bound for the \(y\)-variation.

$$\begin{aligned} \mathbf{E}&\left[|\alpha _{\varepsilon , t}^{\prime }\left(y,A_k^j\right)-\alpha _{\varepsilon , t}^{\prime }\left(\tilde{y},A_k^j\right)|^n\right] \nonumber \\&\le C|y-\tilde{y}|^{n\lambda } \int \limits _{\mathbb{R }^n} \frac{\prod _{k=1}^n|p_k| ^{1+\lambda }d\overrightarrow{p}}{\prod _{k=1}^n \left(1+|u_k|^{1/H}\right)\prod _{k=1}^n \left(1+|u^{\prime }_k|^{1/H}\right)} \nonumber \\&\le C |y-\tilde{y}|^{n\lambda }\Big | \! \Big | \frac{\prod _{k=1}^n |p_k|^{(1+ \lambda )/2}}{\prod _{k=1}^n \left(1+|u_k|^{1/H}\right)} \Big | \! \Big |_2 \Big | \! \Big | \frac{\prod _{k=1}^n |p_k|^{(1+\lambda )/2}}{\prod _{k=1}^n \left(1+|u_k|^{1/H}\right)} \Big | \! \Big |_2 \nonumber \\&\le C |y\!-\!\tilde{y}|^{n\lambda }\Big | \! \Big | \prod _{k\!=\!1}^n\frac{1\!+\!|u_k |^{(1\!+\!\lambda )/2}+|u_k|^{(1\!+\!\lambda )}}{\left(1\!+\!|u_k|^{1/H}\right)} \Big | \! \Big |_2 \Big | \! \Big | \prod _{k\!=\!1}^n \frac{ 1\!+\!|u^{\prime }_k|^{(1\!+\!\lambda )/2}\!+\!|u^{\prime }_k|^{(1\!+\!\lambda )}}{\left(1\!+\!|u_k|^{1/H}\right)} \Big | \! \Big |_2.\nonumber \\ \end{aligned}$$
(3.6)

Similar arguments hold for the \(\varepsilon \) and \(t\) variation.

Another possible approach to proving continuity of \(\hat{\alpha }_{t}^{\prime }(y) - t \mathbf{E}[\hat{\alpha }_{t}^{\prime }(y)]\) is as follows. In Theorem 1.2, the expectation in (2.8) depends upon the ordering of the \(s_k\)’s and \(r_k\)’s. It may be possible to do a more careful analysis of the different possible orderings to handle \(\hat{\alpha }_{t}^{\prime }(y) - t \mathbf{E}[\hat{\alpha }_{t}^{\prime }(y)]\). Of particular interest are the configurations with isolated intervals, that is, values \(k^{\prime }\) for which the interval \([r_{k^{\prime }},s_{k^{\prime }}]\) contains no other \(r_k\) or \(s_k\). For instance, in Figure 1, \(p_2\), and \(p_6\) correspond to isolated intervals, while no others do. It turns out that configurations of \(\{r_1,s_1, \ldots ,r_n,s_n\}\) which have no isolated intervals are still amenable to methods used to prove Theorem 1.2, except that replacing \(m_j\) by 2 in (2.16) and (2.23) no longer results in a convergent integral; however, the arguments of [10] can be applied to this difficulty to get appropriate bounds of the form (2.7) and (2.17). The difficulty, then, lies with the isolated interval case, as can also be seen in several other instances in which this general method has been applied (e.g., [10] and [14]). If isolated intervals are present, then essentially the task is to “remove the isolated intervals,” that is, to integrate out the variables corresponding to isolated intervals to reduce a configuration to a smaller one. For the unnormalized process \(\hat{\alpha }_{t}^{\prime }\), this is possible for sets of the form

$$\begin{aligned} A:= \{0 < r < s-\kappa < t-\kappa \} \ \ \text{ for} 0<\kappa <t. \end{aligned}$$

One approach might be to show that the renormalization, i.e., the subtraction of the term \(t \mathbf{E}[\hat{\alpha }_{t}^{\prime }(y)]\), cancels with integrals over configurations with isolated intervals.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Jung, P., Markowsky, G. Hölder Continuity and Occupation-Time Formulas for fBm Self-Intersection Local Time and Its Derivative. J Theor Probab 28, 299–312 (2015). https://doi.org/10.1007/s10959-012-0474-8

Download citation

Keywords

  • Intersection local time
  • Fractional Brownian motion
  • Occupation-time formula

Mathematics Subject Classification (2010)

  • 60G22
  • 60J55