Sufficient conditions in terms of interpolation variances are given for a Gaussian process to have a jointly continuous local time. In the stationary case these conditions can be verified in terms of the spectral density and are seen to be within logarithmic factors of the best possible conditions. A bound for the modulus of continuity in the space variable is also obtained.
Jack Cuzick. Johannes P. DuPreez. "Joint Continuity of Gaussian Local Times." Ann. Probab. 10 (3) 810 - 817, August, 1982. https://doi.org/10.1214/aop/1176993789