By Peres Y.

Those notes list lectures I gave on the data 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 through 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 college in Jyvaskyla, August 1999.

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

**Sample text**

Ii) If EX = 0 and EX 2 < ∞, then τ can be chosen to have finite mean. Only part (ii) of the theorem is useful. Proof. (i) Pick X according to its distribution. Define τ = min{t : B(t) = X}. s. s (ii) Let X have distribution ν on R. , ν({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}).

For the lower bound, fix q > 1. In order to use the Borel-Cantelli lemma in the other direction, we need to create a sequence of independent events. 5 for large x: P(Z > x) ≥ ce−x x 2 /2 . Using this estimate we get P(Dn) = P Z ≥ and therefore ≥c q n − q n−1 n P(Dn ) e− log log(q −q ) ce− log(n log q) c ≥ > n n−1 n log n 2log log(q − q ) 2 log(n log q) n ψ(q n − q n−1 ) n−1 = ∞. Thus for infinitely many n B(q n ) ≥ B(q n−1 ) + ψ(q n − q n−1 ) ≥ −2ψ(q n−1 ) + ψ(q n − q n−1 ) where the second inequality follows from applying the previously proven upper bound to −B(q n−1 ).

Also, ∞ ∞ ∞ µ(Sk (x))2kα ≤ C k=1 |22−k |β 2kα = C k=1 2k(α−β), k=1 2β where C = 2 C. Since β > α, we have ∞ Eα (µ) ≤ C 2k(α−β) < ∞, k=1 which proves the theorem. Definition. The α-capacity of a set K, denoted Capα (K), is inf Eα (µ) µ −1 , where the infimum is over all Borel probability measures supported on K. If Eα(µ) = ∞ for all such µ, then we say Capα (K) = 0. 3 (McKean, 1955). Let B denote Brownian motion in Rd . Let A ⊂ [0, ∞) be a closed set such that dimH (A) ≤ d/2. Then, almost surely dimH B(A) = 2 dimH (A).