By Peres Y.

Those notes list lectures I gave on the statistics division, college 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 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 university in Jyvaskyla, August 1999.

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**Additional resources for An invitation to sample paths of Brownian motion**

**Example text**

2. Let u be measurable and locally bounded in D. 1) σd−1 (S(x, r)) S(x,r) where S(x, r) = {y : |y − x| = r}, and σd−1 is the (d − 1)-dimensional Hausdorff measure. Proof. Assume u is harmonic. Define: S(x,r) u(y)dσd−1(y) = Ψ(r)r d−1 . We will show that Ψ is constant. Indeed, for any R > 0, Rd Ld (B(x, 1))u(x) = Ld (B(x, R))u(x) = R u(y)dy = B(x,R) Ψ(r)r d−1 dr. t. R to obtain: dLd (B(x, 1))u(x) = Ψ(R). and therefore Ψ(R) is constant. 1) holds. 1) implies that u(x) = Ld (B(x, r))−1 B(x,r) u(y)dy by Fubini’s Theorem.

BROWNIAN MOTION Proof. The upper bound follows from the upper bound for continuous time. To prove the lower bound, we might run into the problem that λn and q n may not be close for large n; we have to exclude the possibility that λn is a sequence of times where the value of Brownian motion is too small. To get around this problem define Dk∗ = Dk ∩ min q k ≤t≤q k+1 B(t) − B(q k ) ≥ − q k def = D k ∩ Ωk Note that Dk and Ωk are independent events. Moreover, by scaling, P(Ωk ) is a constant cq > 0 that does not depend on k.

However, it is possible to define a more sophisticated ϕ-Hausdorff measure for which, with probability one, 0 < Hϕ (ZB ) < ∞. Such a function ϕ is called an exact Hausdorff measure function for ZB . 12. Harris’ Inequality and its consequences We begin this section by proving Harris’ inequality. 1 (Harris’ inequality). Suppose that µ1 , . . , µd are Borel probability measures on R and µ = µ1 × µ2 × . . × µd . Let f, g : Rd → R be measurable functions that are nondecreasing in each coordinate. 1) Rd provided the above integrals are well-defined.

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

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