By Peres Y.

Those notes list lectures I gave on the facts 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 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 season tuition in Jyvaskyla, August 1999.

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**Example text**

2). Let f˜ ∈ C[0, ˜ − ψ(f )| < ε. That is, ψ is continuous for every t, f (t) − δ ≤ f˜(t) ≤ f (t) + δ. Hence, |ψ(f) at f . Since the last zero of a Brownian path on [0, 1] almost surely is strictly less than 1, and is an accumulation point of zeroes from the left, the Brownian path almost surely has the property that f has. Hence, ψ is continuous at almost all Brownian paths. 2. Illustration that shows ψ is continuous at almost all Brownian paths 46 1. 1. Let Stn Fn (t) = √ , 0 ≤ t ≤ 1. n By Skorokhod embedding, there exist stopping times Tk , k = 1, 2, .

Define the stopping time τ−1 = min{k : Sk = −1}. Then pn = P(Sn ≥ 0) − P(Sn ≥ 0, τ−1 < n). Let {Sj∗ } denote the random walk reflected at time τ−1 , that is Sj∗ = Sj Sj∗ = (−1) − (Sj + 1) for j ≤ τ−1 , for j > τ−1 . Note that if τ−1 < n then Sn ≥ 0 if and only if Sn∗ ≤ −2, so pn = P(Sn ≥ 0) − P(Sn∗ ≤ −2). Using symmetry and the reflection principle, we have pn = P(Sn ≥ 0) − P(Sn ≥ 2) = P(Sn ∈ {0, 1}), which means that n pn = P(Sn = 0) = n/2 2−n for n even, n −n pn = P(Sn = 1) = (n−1)/2 2 for n odd.

Thus 1(|y − x| ≤ 2 )K(x, y)dµ(x)dµ(y) EZ 2 ≤ 2µ(B (0)) + 2d−1 +2 1(|y − x| > 2 ) |y| |y − x| − d−2 dµ(x)dµ(y). 6) Since the kernel is infinite on the diagonal, any measure with finite energy must have no atoms. 6) drop out as → 0 by dominated convergence. Thus by the dominated convergence Theorem, lim EZ 2 ≤ 2IK (µ). 7) ↓0 Clearly the hitting probability P(∃t > 0, y ∈ Λ : Wt ∈ B (y)) is at least P(Z > 0) ≥ (EZ )2 = (EZ 2 )−1 . EZ 2 Transience of Brownian motion implies that if the Brownian path visits every -neighborhood of the compact set Λ then it almost surely intersects Λ itself.