Derivatives of local times for some Gaussian fields

Abstract

In this article, we consider derivatives of local time for a (2, d)-Gaussian fieldwhere X H1 and X H2 are two independent processes from a class of d-dimensional centered Gaussian processes satisfying certain local nondeterminism property. We first give a condition for existence of derivatives of the local time. Then, under this condition, we show that derivatives of the local time are Hölder continuous in both time and space variables. Moreover, under some additional assumptions, we show that this condition is also necessary for existence of derivatives of the local time at the origin.

Existence, renormalization, and regularity properties of higher order derivatives of self-intersection local time of fractional Brownian motion

“…The derivative of self-intersection local time of Brownian motion was also introduced by Rosen, in [32], and has blossomed into a research topic in its own right ( [1,20,27,28,33,34,35,36,37,38,39,40,41,42,43,44,45,46]). The original motivation discussed by Rosen was the Tanaka-style formula…”

Derivatives of local times for some Gaussian fields II

“…In this paper, when the location is not the origin, we will show that, if the Gaussian processes in [3] satisfying some additional properties, then the sufficient condition…”

“…Remark 1.4 For independent Gaussian processes X H 1 and X H 2 in G d L as defined in [3], if they also satisfy the assumptions in Theorem 1.2, then L (k) (T, 0) exists in L 2 if and only if…”

“…Local times for Gaussian processes or fields are important in the probability theory. Recently, their derivatives received much attention, see, for example, [5,3] and references therein. In [3], we consider derivatives of local time for a (2, d)-Gaussian field…”

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

“…where X H 1 and X H 2 are two independent Rosenblatt processes. For fBm or more general Gaussian case, Guo et al [4] and Hong and Xu [5,6] study the existence and Hölder continuity. In this paper, we have the results of Rosenblatt case.…”