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When random walk takes the steps at random times, and in that case the position X_t is defined as the continuum of times t≥0, isn't this concept/phenomenon/rule is Brownian motion (Weiner process)? At http://en.wikipedia.org/wiki/Wiener_process in section "Characterizations of the Wiener process" denote it as "W_t", does it tell the position at time "t" just like we find the position in random walks? In the same section, I don't understand the second property, particularly "t→W_t" , what is this?

Thanks in advance

Note: The text says that the mathematical treatment of Brownian motion is called Weiner process, therefore I am thinking that both Brownian motion & Weiner are same, am I correct?