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This is the Exam of Statistical Science which includes Stochastic Differential Equation, Brownian Motion, Solution, Measurable Function, Markov Process, Starting, Bounded Functions, Local Martingale, First Time etc. Key important points are: Stochastic Loewner Evolutions, Conformal Isomorphism, Compact, Set, Compact, Proof, Transform, Relation, Loewner , Distribution
Typology: Exams
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Monday 9 June 2008 1.30 to 3.
Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.
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1 (a) Let K be a compact H-hull and set H = H \ K. Show that there exists a unique conformal isomorphism gK : H → H such that gK (z) − z → 0 as z → ∞.
(b) Suppose that K 0 is a compact H-hull and gK 0 (z) = z + z−^1 for all z ∈ H \ K 0. Identify K 0 and find hcap(K 0 ).
(c) Suppose that |z| ≤ 1 for all z ∈ K. Show that, for all x ∈ R with |x| > 1,
1 − x−^2 ≤ g′ K (x) ≤ 1.
[In (c), you may use without proof any result from the course.]
2 (a) Let (γt)t≥ 0 be an SLE(κ), for some κ ∈ [0, ∞). Explain the relation to (γt)t≥ 0 of the associated Loewner flow (gt)t≥ 0 and transform (ξt)t≥ 0.
(b) Fix s ≥ 0 and define for t ≥ 0
¯γt = gs(γs+t), ˜γt = ¯γt − ξs.
What is the Loewner transform of (¯γt)t≥ 0? What is the distribution of (˜γt)t≥ 0? Justify your answers.
(c) Suppose now that κ ∈ (0, 4]. Show that, almost surely, (γt)t≥ 0 is a simple curve. [You may assume without proof that, almost surely, Im(γt) > 0 for all t > 0.]
3 (a) Let γ be an SLE(8/3). Let U be a simply connected domain in the upper half-plane H, which is a neighbourhood of both 0 and ∞. Denote by Φ the unique conformal isomorphism U → H such that Φ(z)−z → 0 as z → ∞. Set Kt = {γs : 0 < s ≤ t} and T = inf{t ≥ 0 : γt ∈/ U }. Define, for t < T , K t∗ = {Φ(γs) : 0 < s ≤ t}, Φt = gK t∗ ◦ Φ ◦ g− K^1 t ,
where, for K a compact H-hull, gK : (H \ K) → H is the unique conformal isomorphism such that gK (z) − z → 0 as z → ∞. Set Σt = Φ′ t(ξt),
where ξ is the Loewner transform of γ. Show that a suitably chosen function of the process Σ is a local martingale.
(b) Hence, show that
P(γt ∈ U for all t ≥ 0) = Φ′(0)^5 /^8.
[You may assume without proof any standard identities of the classical Loewner theory, or for the Brownian excursion. You may also assume that Σt → (^1) {T =∞} as t ↑ T , almost surely.]
Paper 36
