  We study a continuous pathwise local time of order p for continuous functions
with finite p-th variation along a sequence of time partitions, for even
integers p >= 2. With this notion, we establish a Tanaka-type change of
variable formula, as well as Tanaka-Meyer formulae. We also derive some
identities involving this high-order pathwise local time, each of which
generalizes a corresponding identity from semimartingale theory. We then use
collision local times between multiple functions of arbitrary regularity, to
study the dynamics of ranked continuous functions of arbitrary regularity. We
present also another definition of pathwise local time which is more natural
for fractional Brownian Motions, and give a connection with the previous notion
of local time.
