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.

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.