Analytical aspects of Liouville-type equations with singular by Tarantello G.

By Tarantello G.

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7, and the latter is more interesting. Let V. be a uniform random V the ba(n)'th generation vertex of CBP(n). Let before V, and let be the total number of descendants of Then from Theorem 5 (for re-rooted CBP(n)) Z(b). In [2] section 7 the global construction was used to prove Lemma 10 (Z(b), b > 0) is the positive stable (1/2) process, that is Eexp(-9Z(b)) = exp(-b 29) Z(b) -A b 2Z(1). It is convenient to record an easy calculation here: jb(b - s)Z(ds) d (4/9)b3Z(1). (44) One can alternatively obtain Lemma 10 from the BES(3) representation.

28) Aldous: The continuum random tree II 47 have some relevance to interesting questions about CBP(n). Here is one example: others are in the next section. Odlyzko and Wilf [38] were interested in the maximal height profile Hn = maxH(j) 7 for CBP(n). *. In view of (28), Conjecture 4 would imply n- 1/2H. d and suggest the result for means n-1/2EHn -, a x/2. Finally, one could consider the sum >J i j H(j) of heights of all n vertices of CBP(n). Corollary 3 implies Corollary 9 n-3/2 > jH (j) - 2Q-1I i where I = f1 J0 W,ds.

1991), Local time and Tanaka formulae for super Brownian motion and super stable processes, Stochastic Processes and Their Applications, to appear. J. S. (1991), Intersection local times of all orders for Brownian and stable density processes-construction, renormalization and limit laws, Ann. Probability, to appear. [4] Dawson, D. (1978), Geostochastic calculus, Canadian J. Statistics, 6, 143-168. [5] Dawson, D. (1986), Measure-valued processes: construction, qualitative behaviour and stochastic geometry, Proc.

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