By Peres Y.

Those notes checklist lectures I gave on the records division, collage of California, Berkeley in Spring 1998. i'm thankful to the scholars who attended the path and wrote the 1st draft of the notes: Diego Garcia, Yoram Gat, Diogo A. Gomes, Charles Holton, Frederic Latremoliere, Wei Li, Ben Morris, Jason Schweinsberg, Balint Virag, Ye Xia and Xiaowen Zhou. The draft used to be edited by means of Balint Virag, Elchanan Mossel, Serban Nacu and Yimin Xiao. I thank Pertti Mattila for the invitation to lecture in this fabric on the joint summer season college in Jyvaskyla, August 1999.

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**Extra resources for An invitation to sample paths of Brownian motion**

**Example text**

0}) = 0. For, suppose ν({0}) > 0. Write ν = ν({0})δ0 + (1 − ν({0})˜ ν , where the distribution ν˜ has no mass on {0}. Let stopping time τ˜ be the solution of the problem for the distribution ν˜. The solution for the distribution ν is, τ= τ˜ with probability 1 − ν({0}) 0 with probability ν({0}). 40 1. BROWNIAN MOTION Then, Eτ = (1 − ν({0}))E˜ τ < ∞ and B(τ ) has distribution ν. From now on, we assume ν({0}) = 0. ¿From EX = 0 it follows that: ∞ def M = x dν = − 0 0 x dν. −∞ Let φ : R −→ R be a non-negative measurable function.

For the second term we just use the fact that ψ(t)/t is decreasing in t. s. and letting q ↑ ∞ concludes the proof of the upper bound. 5. s. s. , then lim sup n→∞ 38 1. BROWNIAN MOTION Proof. The upper bound follows from the upper bound for continuous time. To prove the lower bound, we might run into the problem that λn and q n may not be close for large n; we have to exclude the possibility that λn is a sequence of times where the value of Brownian motion is too small. To get around this problem define Dk∗ = Dk ∩ min q k ≤t≤q k+1 B(t) − B(q k ) ≥ − q k def = D k ∩ Ωk Note that Dk and Ωk are independent events.

2 illustrates both the Skorokhod’s stopping rule and the Root’s stopping rule. In Root’s stopping rule, the two dimensional set A consists of four horizontal lines represented by {(x, y) : x ≥ M, |y| = 1} ∪ {(x, y) : x ≥ 0, |y| = 2}, for some M > 0. This is intuitively clear by the following argument. Let M approache 0. The Brownian motion takes value of 1 or −1, each with probability 1/2, at the first time the Brownian graph hits the set A. Let M approache ∞. The Brownian motion takes value of 2 or −2, each with probability 1/2, at the first time the Brownian graph hits the set A.

### An invitation to sample paths of Brownian motion by Peres Y.

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