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

By Peres Y.

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

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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. (2) G(x, y) = G(y, x) = G(y − x, 0). (3) G(x, 0) = cd |x|2−d where cd = Γ(d/2 − 1)/(2π d/2). Proof. Facts 1. and 2. are immediate. , note that: ∞ G(x, 0) = (2πt)−d/2 exp − 0 Substituting s = |x|2 2t |x|2 dt. 2t , we obtain: ∞ G(x, 0) = −d/2 π|x|2 −d/2 −s |x|2 2−d π e ds = |x| ) s 2s2 2 ( 0 ∞ e−s s 2 −2 ds.

Is there an infinite cylinder avoided by W ? 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?

1). 5). Stop when the Brownian motion reaches either Z or Y for 15. SKOROKHOD’S REPRESENTATION 41 the first time. Notice that τ is a stopping time with respect to the Brownian filtration Ft = σ{{B(s)}s≤t , Z, Y }. d Next, we will show B(τ ) = X. Indeed, for any bounded measurable function φ: Eφ(B(τ )) = E[E[φ(B(τZ,Y ))|Z, Y ]] = E[ {Z,Y } φ dµZ,Y ] = Eφ(X). 6). The expectation of τ can be computed similarly: Eτ = E[E[τZ,Y |Z, Y ]] = E[ {Z,Y } x2 dµZ,Y ] = x2 dν(x). 6), by letting φ(x) = x2 . 1.

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