A Williams decomposition for spatially dependent superprocesses

Abstract

43 pagesInternational audienceWe present a genealogy for superprocesses with a non-homogeneous quadratic branching mechanism, relying on a weighted version of the superprocess and a Girsanov theorem. We then decompose this genealogy with respect to the last individual alive (William's decomposition). Letting the extinction time tend to infinity, we get the Q-process by looking at the superprocess from the root, and define another process by looking from the top. Examples including the multitype Feller diff usion and the superdiffusion are provided

Williams decomposition for superprocesses

“…Using this, one can check that Assumption (H2) is satisfied. This example shows that our result covers Delmas and Hénard [4,Corollary 4.14].…”

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We decompose the genealogy of a general superprocess with spatially dependent branching mechanism with respect to the last individual alive (Williams decomposition). This is a generalization of the main result of Delmas and Hénard [4] where only superprocesses with spatially dependent quadratic branching mechanism were considered. As an application of the Williams decomposition, we prove that, for some superprocesses, the normalized total mass will converge to a point mass at its extinction time.

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“…More precisely, we give a spinal decomposition of X involving the ancestral lineage of the last individual alive, conditioned on H = h with h > 0 being a constant. This decomposition is called a Williams decomposition, in analogy with the terminology of Delmas and Hénard [4]. For a superprocess with spatially independent branching mechanism, the spatial motion is independent of the genealogical structure.…”

“…In [4], the Williams decomposition was established for superprocesses with spatially dependent quadratic branching mechanism by using two transformations to change the branching mechanism Ψ(x, z) to a spatially independent one, say ψ 0 , and then using the genealogy of superprocesses with branching mechanism ψ 0 given by the Brownian snake. As mentioned in [4], the drawback of the approach in [4] is that one has to restrict to quadratic branching mechanisms with bounded and smooth parameters.…”

Critical Multi-type Galton–Watson Trees Conditioned to be Large

“…When scaling limits of multi-type GW tree are considered, one obtains as a limit a continuous GW tree, see Miermont [17] or Gorostiza and Lopez-Mimbela [16] (when the probability to give birth to different types goes down to 0). In this latter case see Delmas and Hénard [6] for the limit on the conditioned random tree to have a large height.…”