I would like to ask for a more rigorous statement and proof of Lemma on page 5 of this paper. In essence, it states that two distinct sample paths of a Brownian motion does not strictly cross (meaning once they intersect at some time, they merge together from then on) with probability $1$. They claim this stems from the Markov property of the Brownian motion which stipulates that a path is uniquely determined by its initial position. This seems strange to me. All the sample path starts from $0$ at time $0$. If the statement is true, then there would have been only one path and the process would have been deterministic. Also Markov property in essence states that the conditional probability depends not on the historical but the current state. It does not seem to say two paths can not start from the same point.
Can someone resolve this confusion?