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Various paths properties of a stochastic process are obtained under mild conditions which allow for the integrability of the characteristic function of its increments and for the dependence among them. The main assumption is closely related to the notion of local asymptotic self-similarity. New results are obtained for the class of multifractional random processes .

We establish estimates for the local and uniform moduli of continuity of the local time of multifractional Brownian motion, B H = (B H(t) (t), t ∈ R + ). An analogue of Chung's law of the iterated logarithm is studied for B H and used to obtain the pointwise Hölder exponent of the local time. A kind of local asymptotic self-similarity is proved to be satisfied by the local time of B H .

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