By Jerry Marsden

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2 |y|≤t−1/2 e−1/2|y| dy ∀t). Theorem . (Strong Markov Property of Brownian Motion). e. e. Let Yt = Xτ+t − Xτ . Then 1. P[(Yt1 ∈ A1 , . . , Ytk ∈ Ak ) ∩ A] = P(Xt1 ∈ A1 , . . Xtk ∈ Ak ) · P(A), ∀ A ∈ Fτ and for every Ai Borel in Rd . Consequently, 2. (Yt ) is a Brownian motion. 3. (Yt ) is independent of Fτ . The assertion is that a Brownian motion starts afresh at every stopping time. Proof. Step 1. Let τ take only countably many values, say s1 , s2 , s3 . .. Put E j = τ−1 {s j }. Then each E j is Fτ -measure and Ω= ∞ j=1 E j , E j ∩ Ei = ∅ j i.

4 2. Let {Xt } be a d-dimensional Brownian motion, G any closed set in Rd . Define τ(w) = inf{t : w(t) ∈ G}. This is a generalization of Example 1. To see that τ is a stopping time use {τ ≤ s} = ∞ lim {w : w(θ) ∈ Gn }, θ∈[0,s] n=1 θ rational where Gn = x ∈ Rd : d(x, G) ≤ 1 . n 3. Let (Xt ) be a d-dimensional Brownian motion, C and D disjoint closed sets in Rd . Define τ(w) = inf{t; w(t) ∈ C and for some s ≤ t, w(s) ∈ D}. τ(w) is the first time that w hits C after visiting D. 5. Generalised Brownian Motion LET Ω BE ANY space, F a σ-field and (Ft ) an increasing family of 31 sub σ-fields such that σ(∪Ft ) = F .

Note that P x (|Xr + · · · + X1 | ≥ δ) = P(|X(t′ ) − X(t′′ )| ≥ δ) for some t′ , t′′ in F (*) E(|X(t′ ) − X(t′′ )|4 ) (see Tchebyshey’s inequality in Appendix) δ4 C ′ (t′′ − t′ ) (C ′′ = constant) ≤ δ4 C ′ |I|2 ≤ . δ4 ≤ Therefore ǫ ≤ C ′ |I|2 . Now δ4 P x ( sup |X(t) − X(σ)| ≥ 4δ) t,σ∈P P x ( sup |X(ti ) − X(t1 )| ≥ 2δ) 1≤i≤k = P x ( sup |X1 + · · · + X j | ≥ 2) ≤ 2ǫ (by previous lemma) i≤ j≤k−1 2C ′ |I|2 C|I|2 = 4 . δ4 δ 25 4. Construction of Wiener Measure 26 Exercise 5. Verify (∗). (Hint: Use the density function obtained in Exercise 2(c) to evaluate the expectation and go over to “popular” coordinates.