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

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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 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 college in Jyvaskyla, August 1999.

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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). Remark. 3 requires A to be fixed. If we allow a random A depending on the Brownian path, then the conclusion still holds if d ≥ 2. However, for d = 1, suppose A = ZB = {t : B1 (t) = 0}.

Skorokhod’s representation for a sequence of random variables. Let {Xi}i≥1 be independent random variables with mean 0 and finite variances. Let τ1 be a d stopping time with Eτ1 = EX12 and B(τ1 ) = X1 . {B(τ1 + t) − B(τ1 )}t≥0 is again a Brownian d motion. Then, we can find a stopping time τ2 with Eτ2 = EX22 , and B(τ1 +τ2 )−B(τ1 ) = X2 and is independent of Fτ1 . Repeat the procedure for τ3 , τ4 · · · , etc. Define T1 = τ1 , and d Tn = τ1 + τ2 + · · · + τn . Then, B(Tk + τk+1 ) − B(Tk ) = Xk+1 and is independent of FTk .

4 (The Law of the Iterated Logarithm). s. Remark. s. ψ(t) Khinchin proved the Law of Iterated Logarithm for simple random walk, Kolmogorov for other walks, and L´evy for Brownian motion. The proof for general random walks is much simpler through Brownian motion than directly. Proof. The main idea is to scale by a geometric sequence. We will first prove the upper bound. Fix > 0 and q > 1. Let An = max B(t) ≥ (1 + )ψ(q n) . 6 the maximum of Brownian motion up to a fixed time t has the same distribution as |B(t)|.

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