Download PDF by Peres Y.: An invitation to sample paths of Brownian motion

By Peres Y.

Those notes list lectures I gave on the facts 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 by way of 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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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. Remark. Here is an equivalent definition for harmonicity. u is harmonic if u is contin∂ 2u uous, twice differentiable, and ∆u = i (∂x 2 = 0. i) 48 1. 3. ∞ G(x, y) = p(x, y, t)dt, x, y ∈ Rd 0 is the Green function in Rd , where p(x, y, t) is the Brownian transition density function, 2 p(x, y, t) = (2πt)−d/2 exp − |x−y| . 4. The Green function G satisfies: (1) G(x, y) is finite iff x = y and d > 2.

1) can be made arbitrarily small by first taking n large and then picking h > 0 sufficiently small. 13. 3. For any random walk {Sj } on the line, P(S0 , . . , Sn has a point of increase) ≤ 2 n k=0 pk pn−k n/2 2 k=0 pk . 2) Proof. The idea is simple. 3), and given that there is at least one such point, the expected number is bounded below by the denominator; the ratio of these expectations bounds the required probability. To carry this out, denote by In (k) the event that k is a point of increase for S0 , S1 , .

It is therefore better to estimate hitting probabilities by a capacity function with respect to a scale-invariant modification of the Green kernel, called the Martin kernel: K(x, y) = G(x, y) |y|d−2 = G(0, y) |x − y|d−2 for x = y in Rd , and K(x, x) = ∞. The following theorem shows that Martin capacity is indeed a good estimate of the hitting probability. 3 (Benjamini, Pemantle, Peres 1995). Let Λ be any closed set in Rd , d ≥ 3. 2) CapK Λ ≤ P(∃t > 0 : W (t) ∈ Λ) ≤ CapK (Λ) 2 Here −1 CapK (Λ) = inf µ(Λ)=1 K(x, y)dµ(x)dµ(y) Λ .

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