Law of large numbers for a class of superdiffusions
By János Engländer, A. Winter. 2005
57 citations recorded by the source corpus.
Abstract Pinsky [R.G. Pinsky, Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions, Ann. Probab. 24 (1) 237–267] proved that the finite mass superdiffusion X corresponding to the semilinear operator L u + β u − α u 2 exhibits local extinction if and only if λ c ⩽ 0 , where λ c : = λ c ( L + β ) is the generalized principal eigenvalue of L + β on R d . For the case when λ c > 0 , it has been shown in Englander and Turaev [J. Englander, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] that in law the superdiffusion locally behaves like exp [ t λ c ] times a non-negative non-degenerate random variable, provided that the operator L + β − λ c satisfies a certain spectral condition (‘product-criticality’), and that α and μ = X 0 are ‘not too large’. In this article we will prove that the convergence in law used in the formulation in [J. Englander, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] can actually be replaced by convergence in probability. Furthermore, instead of R d we will consider a general Euclidean domain D ⊆ R d . As far as the proof of our main theorem is concerned, the heavy analytic method of [J. Englander, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] is replaced by a different, simpler and more probabilistic one. We introduce a space–time weighted superprocess (H-transformed superprocess) and use it in the proof along with some elementary probabilistic arguments.
Subfields: Computational Theory and Mathematics, Finance, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/fd4aa3120d7ff95d7cb51f64923ee705d629c247
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Citations in the corpus
- Limit theorems for Markov processes indexed by continuous time Galton--Watson trees (2009, background): “Section 2.2 is devoted to the definition of tree indexed Markov processes.”
- Limit theorems for some branching measure-valued processes (2011, methodology): “a. The key point of our weighted many-to-one formula is an h−transform (Girsanov type transformation) of the Feynman-Kac semigroup as in [39]. This transformation is usual in the superprocesses study [16]. Remark 3.6 (Galton-Watson tree and Malthus parameter). If rand pare constant, then V ≡ 1 is an”