LOCAL TIME AS AN ELEMENT OF THE SOBOLEV SPACE

ALEXEY V. RUDENKO

Theory of Stochastic Processes Vol. 13 (29), no. 3, 2007, pp. 65–79

For a centered Gaussian random field taking its values in $\mathbb{R}^d$, we investigate the existence of a local time as a generalized functional, i.e an element of some Sobolev space. We give the sufficient condition for such an existence in terms of the field
covariation and apply it in several examples: the self-intersection local time for a fractional Brownian motion and the intersection local time for two Brownian motions.

Full version