Local Behaviour of First Passage Probabilities - MIMS - The ...

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After reading o¤ the asymptotic behaviour <strong>of</strong> the …rst term in (64) from(56), the pro<strong>of</strong> is now completed by using (65), (66), (67), and the followingresult.Lemma 20 For x; y 0 and > 0 the following identity holdsZZU(x z)dzV (dw z) + U(x dz)V (w z)dw0z

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An asymptotic lower bound is given byr>nf(0)d(n)bncXr=1Z0z

After reading o¤ the asymptotic behaviour <strong>of</strong> the …rst term in (64) from(56), the pro<strong>of</strong> is now completed by using (65), (66), (67), and the followingresult.Lemma 20 For x; y 0 and > 0 the following identity holdsZZU(x z)dzV (dw z) + U(x dz)V (w z)dw0z

Page 1: Local Behaviour of First Passage Pr
Page 5: From this we get immediately a stre
Page 8 and 9: From this we can deduce the followi
Page 10 and 11: in t; of the LHS of (34) is the sam
Page 12 and 13: where lim n; () = 1 is shorthand fo
Page 14 and 15: (iii) In this case it is P 2 that d
Page 16 and 17: so one part of (45) will follow if
Page 18 and 19: 5.3 Proof when x=c n ! 1Proof. This
Page 20 and 21: 6 The non-lattice caseWe indicate h
Page 24 and 25: References[1] V. I. Afanasyev, C. B