On sojourn times for functions and stochastic processes

For processes with $d$-dimensional ($d>1$) parameter and with realizations in the class Lip 1, sufficient conditions for the existence of the density of a sojourn time are given. If realizations of the process are in the class $C^d$ ($[0,1]^d$), these conditions are also necessary. For analogs of the Brownian motion with $d$-dimensional parameter, estimates for the smoothness of the sojourn time density are obtained.