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By Lapeyre B., Delmas J.-F.

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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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