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Understanding "W_t returns infinitely oftern to 0..."

R

Rose95

Member
Hi there, one of the properties of the Wiener process W_t is that it returns infinitely often to 0, or indeed to any other level. What does this actually mean?
 
Staring at zero, a Wiener process wobbles away creating a continuous sample path. If you were to zoom in around the origin and observe what the process is doing, it would still look like noisy Brownian motion which never calms down, no matter what magnification you used. Because you can keep increasing the magnification and keep on zooming into the origin, the Wiener process will keep crossing the x-axis; returning infinitely often to zero.
The same argument can be applied to any value the process has already attained.
Hope that helps.
 
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