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By Peres Y.

Those notes checklist lectures I gave on the records 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 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 season institution in Jyvaskyla, August 1999.

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Or equivalently, what is the value of P0 {All orthogonal projections of W are neighborhood recurrent}? , so the probability in the last display vanishes. This is due to Adelman, Burdzy and Pemantle (1998): Sets avoided by Brownian motion. Ann. Probab. 26, 429–464. 21. Capacity and harmonic functions In this section we will characterize the sets that BM hits, and give bounds on the hitting probabilities in terms of capacity. The central question of this section is the following: which sets Λ ⊂ Rd does Brownian motion hit with positive probability?

N . By Harris’ Inequality, Rn f g dµ ≥ f dµ Rn g dµ Rn = p2n . Also, f g dµ = Rn Rn 1{ for all j, x1 + . . + xj ≥ 0 and xj+1 + . . + xn ≥ 0} dµ = P(0 ≤ Sj ≤ Sn for all 1 ≤ j ≤ n), which proves the lower bound. 13. Points of increase for random walks and Brownian motion The material in this section has been taken, with minor modifications, from Peres (1996). A real-valued function f has a global point of increase in the interval (a, b) if there is a point t0 ∈ (a, b) such that f (t) ≤ f (t0 ) for all t ∈ (a, t0 ) and f (t0 ) ≤ f (t) for all t ∈ (t0 , b).

Then, B(Tk + τk+1 ) − B(Tk ) = Xk+1 and is independent of FTk . We get, d B(Tn ) = X1 + X2 + · · · + Xn n ETn = n Eτi = i=1 EXi2 i=1 This is a very useful formulation. d. d. By the Strong Law of Large Numbers, Tnn −→ Eτ1 = EX12 almost surely, as n −→ ∞. Let Sn = ni=1 Xi = B(Tn ). 5 of the Law of Iterated Logarithm (LIL) for the Brownian motion, we have, lim sup √ n→∞ Sn =1 2n log log n EX12 This was first proved by Strassen (1964). 16. d. random variables with mean 0 and finite variances. By normalization, we can assume the variance Var(Xi ) = 1, for all i.

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