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.

On the Tanaka formula for the derivative of self-intersection local time of fractional Brownian motion

“…This other version is more natural for an occupation times formula, as well as for differentiation in the space variable y. An interested reader is referred to [JM12] for details. However, as mentioned above, our primary interest here lies in the Tanaka formula (1.21), which does require the kernel to be present.…”

Existence and Hölder continuity conditions for self-intersection local time of Rosenblatt process

“…It was first studied by Rosen in [15], this has aroused the interest of many scholars in this research direction. The corresponding results are not only enriched from one-dimensional to multi-dimensional, but also extend the SLT itself to its derivative, including [2], [7], [8], [9], [10], [12], [13], [16], [22], [24]- [26] and references therein.…”

Renormalized self-intersection local time of bifractional Brownian motion

“…On the other hand, since the work of Varadhan [ 16 ], self-intersection local time, as an important topic of probability theory, has been widely considered and studied in recent years. Especially, when it comes to Brownian motion and fractional Brownian motion, it has been extensively studied, see [ 1 , 2 , 4 , 6 , 10 , 11 , 17 ] and the references therein.…”