By Peres Y.

Those notes checklist lectures I gave on the information division, college 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 was once edited by way 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 time university in Jyvaskyla, August 1999.

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

**Sample text**

2IK (µ) 22. KAUFMAN’S THEOREM ON UNIFORM DIMENSION DOUBLING 57 Since this is true for all probability measures µ on Λ, we get the desired conclusion: 1 P(∃t > 0 : Wt ∈ Λ) ≥ CapK (Λ). 2 Remark. 2) can be an equality– a sphere centered at the origin has hitting probability and capacity both equal to 1. The next exercise shows that the constant 1/2 on the left cannot be increased. 4. Consider the spherical shell ΛR = {x ∈ Rd : 1 ≤ |x| ≤ R}. Show that limR→∞ CapK (ΛR ) = 2. 22. s. s. To see the distinction, consider the zero set of 1 dimensional Brownian motion.

2. 6 P( max B(t) ≥ a) = 2P(B(1) ≥ a). 0≤t≤1 To prove this, let φ : R −→ R be a bounded continuous function. Take the function ˜ 1]. By the ψ(f ) = φ(max[0,1] f ). Then ψ is a bounded and continuous function on C[0, construction of {St }t≥0, we have Stn Stn max1≤k≤n Sk √ Eψ({ √ }0≤t≤1 ) = Eφ( max { √ }) = Eφ( ). 0≤t≤1 n n n Also, Eψ({B(t)}0≤t≤1) = Eφ( max B(t)). 0≤t≤1 Then, by Donsker’s Theorem, Eφ( max1≤k≤n Sk √ ) −→ Eφ( max B(t)). 3. 1) max{1 ≤ k ≤ n : Sk Sk−1 ≤ 0} =⇒ max{0 ≤ t ≤ 1|B(t) = 0} n The left hand side is the last time between 1 to n, scaled by n, that the random walk crosses 0.

2. 6 P( max B(t) ≥ a) = 2P(B(1) ≥ a). 0≤t≤1 To prove this, let φ : R −→ R be a bounded continuous function. Take the function ˜ 1]. By the ψ(f ) = φ(max[0,1] f ). Then ψ is a bounded and continuous function on C[0, construction of {St }t≥0, we have Stn Stn max1≤k≤n Sk √ Eψ({ √ }0≤t≤1 ) = Eφ( max { √ }) = Eφ( ). 0≤t≤1 n n n Also, Eψ({B(t)}0≤t≤1) = Eφ( max B(t)). 0≤t≤1 Then, by Donsker’s Theorem, Eφ( max1≤k≤n Sk √ ) −→ Eφ( max B(t)). 3. 1) max{1 ≤ k ≤ n : Sk Sk−1 ≤ 0} =⇒ max{0 ≤ t ≤ 1|B(t) = 0} n The left hand side is the last time between 1 to n, scaled by n, that the random walk crosses 0.

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

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