By Peres Y.

Those notes list lectures I gave on the information 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 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 time college in Jyvaskyla, August 1999.

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**Extra info for An invitation to sample paths of Brownian motion**

**Example text**

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, . . for some standard B(nt) Brownian motion B such that S(k) = B(Tk ). Define Wn (t) = √n . Note that Wn is also a standard Brownian motion. We will show that for any > 0, as n → ∞, P( sup |Fn − Wn | > ) → 0. 3) → 0. Now |Eψ(Fn ) − Eψ(Wn )| ≤ E|ψ(Fn ) − ψ(Wn )| and the right hand side is bounded above by 2M P (||Wn − Fn || ≥ ) + 2M P (||Wn − Fn || < , |ψ(Wn) − ψ(Fn )| > δ) + δ.

2) can be an equality– a sphere centered at the origin has hitting probability and capacity both equal to 1. The next exercise shows that the constant 1/2 on the left cannot be increased. 4. Consider the spherical shell ΛR = {x ∈ Rd : 1 ≤ |x| ≤ R}. Show that limR→∞ CapK (ΛR ) = 2. 22. s. s. To see the distinction, consider the zero set of 1 dimensional Brownian motion. 1. Consider a cube Q ⊂ Rd centered at a point x and having diameter 2r. Let W be Brownian motion in Rd , with d ≥ 3. Define recursively τ1 = inf{t ≥ 0 : W (t) ∈ Q} τk+1 = inf{t ≥ τk + r 2 : W (t) ∈ Q} with the usual convention that inf ∅ = ∞.

6 We have therefore proved that x → |x|1d−2 is harmonic in Rd \ {0}, d ≥ 3. 2). By the definition of FR (0), this expectation can be written as B(0,R) G(0, x)dx = cd B(0,R) |x|2−d dx = c˜d R 0 r d−1 r 2−d dr = cd R2 , in particular, it is finite. We now wish to show that Brownian motion in Rd , d ≥ 3 is transient. 7. For d ≥ 3 and |x| > r, def hr (x) = Px ∃t ≥ 0 : Wt ∈ B(0, r) = r |x| d−2 . Proof. Recall the definition of Fr (x) Fr (x) = G(x − z, 0)dz. 5) we have Fr (x) = Ld (B(0, r))G(x, 0) = Ld (B(0, r))cd|x|2−d .