| Cut Times for Brownian Motion and Random Walk (1999) | |||||||||||||||
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| A cut time for a Brownian motion or a random walk is a time at which the past and the future of the process to not intersect. In this paper we review the work of Erdos on cut times and discuss more recent work on in this area. 1Erdos and cut times Let B t be a Brownian motion taking values in R d ,andletSn be a simple random walk taking values in Z d . A cut time is a time such that the paths of the process before that time and after that time do not intersect. To be precise, we say that t # (0, 1) is a cut time for Brownian motion on [0, 1] if B r #= B s , 0 # r<t<s# 1, i.e., if B[0,t) # B(t, 1] = #. Similarly m is a cut time for simple random walk on [0,n]if S j #= S k , 0 # j # m<k# n, i.e., if S[0,m] # S[m +1,n]=#. # Research supported by the National Science Foundation 1 The question as to whether a "typical" t is a cut time for Brownian motion was answered by Dvoretzky, Erdos, and Kakutani (referred to below as DEK) in [8]. They showed that if 0 <a ... | |||||||||||||||
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