an:06549832
Zbl 1331.60154
Rudenko, Alexey
Local time for Gaussian processes as an element of Sobolev space
EN
Commun. Stoch. Anal. 3, No. 2, 223-247 (2009).
00353291
2009
j
60J55 60G15 60G22 46N30 46E30 60H07
Summary: We consider local time for a Gaussian process with values in \(\mathbb R^d\). We define it as a limit of the standard approximations in Sobolev space. We also study renormalization of local time, by which we mean the modification of the standard approximations by subtracting a finite number of the terms of its It??-Wiener expansion. We prove that renormalized local time exists and is continuous in Sobolev space under a certain condition on the covariation of the process (the condition is general and includes the non-renormalized local time case). This condition is also necessary for the existence of local time if we consider renormalized local time at zero for a zero-mean Gaussian process. We use our general result to obtain a necessary and sufficient condition for the existence of renormalized local time and self-intersection local time for fractional Brownian motion in \(\mathbb R^d\).