LOCAL TIME FOR GAUSSIAN PROCESSES AS AN ELEMENT OF SOBOLEV SPACE

Abstract

In this paper we consider local time for Gaussian process with values inR d . We define it as a limit of the standart approximations in Sobolev space. We also study renormalization of local time, by which we mean the modification of the standart approximations by subtracting a finite number of the terms of its Ito-Wiener expansion. We prove that renormalized local time exists and continuous in Sobolev space under some condition on the covaria- tion 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 zero-mean Gaussian process. We use our general result to obtain the necessary and sucient con- dition for the existence of renormalized local time and self-intersection local time for fractional Brownian motion in R d . for the existence of local time in certain Sobolev space. This condition also provides the continuity of the local time in Sobolev space under weak convergence of associated measures. We prove that our condition is necessary for the existence of local time at zero in the same Sobolev space. Our approach works for the wide class of Gaussian processes. As an application we consider local time and self-intersection local time (with renormalization) for multidimensional fractional Brownian motion and obtain the conditions on the parameters which are sucient and necessary for the existence of renormalized local time in Sobolev