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A Probabilistic Approach to the Equation Lu=−u2

✍ Scribed by E.B. Dynkin

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
172 KB
Volume
170
Category
Article
ISSN
0022-1236

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✦ Synopsis

Let L be a second order elliptic differential operator and let D be an arbitrary open subset of R d . In we introduced a class U 1 (D) of positive solutions of the equation Lu=&u 2 which is in 1 1 correspondence with a convex class H 1 (D) of positive solutions of the equation Lu=0. In the present paper, we give a probabilistic characterization of U 1 (D) and a probabilistic representation of u # U 1 (D) in terms of a superdiffusion. Similar results are obtained also for a parabolic equation u* +Lu=&u 2 .

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