By Peres Y.

Those notes list lectures I gave on the data division, college of California, Berkeley in Spring 1998. i'm thankful to the scholars who attended the direction 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 via 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 resources for An invitation to sample paths of Brownian motion**

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Both probabilities are invariant under changing the scaling factor r. 2. s. there exists a random variable C(ω) so that for all m and for all cubes Q Q ∈ Dm we have τK+1 = ∞ with K = C(ω)m. [− 12 , 12 ]d . Proof. P(τ Qcm+1 < ∞) ≤ m Q∈Dm d c 2dm θcm m Choose c so that 2 θ < 1. Then by Borel-Cantelli, for all but finitely many m we have τ Qcm+1 +1 = ∞ for all Q ∈ Dm . Finally, we can choose a random C(ω) > c to handle the exceptional cubes. 58 1. 3 (Kaufman’s Uniform Dimension Doubling for d ≥ 3).

Which curves can and can not be covered by a Brownian motion path is, in general, an open question. Also unknown is the minimal Hausdorff dimension of curves in a typical Brownian motion path. 47. 14. Frostman’s Lemma, energy, and dimension doubling In this section, we prove a Lemma due to Frostman (1935) and show how it can be used to prove a converse to the energy theorem. We then show how the energy theorem can be used to deduce a result concerning the Hausdorff dimension of B(K) for closed sets K.

It will be used in the next section. 5. We have p2n ≤ P(0 ≤ Sj ≤ Sn for all 1 ≤ j ≤ n) ≤ p2n/2 . Proof. The two events A = {0 ≤ Sj for all j ≤ n/2} and B = {Sj ≤ Sn for j ≥ n/2} are independent, since A depends only on X1 , . , X maining X n/2 +1 , . . , Xn. Therefore, n/2 and B depends only on the re- P(0 ≤ Sj ≤ Sn ) ≤ P(A ∩ B) = P(A)P(B) ≤ p2n/2 , which proves the upper bound. For the lower bound, we follow Peres (1996) and let f (x1 , . . , xn) = 1 if all the partial sums x1 + . . + xk for k = 1, .