Local times for stochastic processes with multidimensional parameters

A definition of a local time is proposed which is applicable for processes with arbitrary parameter set. The existence of a local time is proved to be equivalent to the absolute continuity of sojourn times with respect to the Borel measure. It is shown that it is still possible to refer to local times as measures on level sets. If the parameter set $T$ is a product $T_1\times T_2$, local times are represented as integrals over $T_1$ of local times for the restrictions of the process considered onto $t_1$-sections of $T$. This representation is used to prove the existence of jointly continuous versions of local times for analogs of the Brownian motion with multidimensional parameter.