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Probabilités de présence dans un processus de branchement spatial markovien. (Presence probability in a spatial Markov branching process). (English) Zbl 0611.60082
Given the point processes $$\{M_{t,x}:$$ $$t\geq 0$$, $$-\infty <x<\infty \}$$, a sequence of spatial branching processes, indexed by a small parameter $$\epsilon$$ is constructed in the following way. Let $$S_{\epsilon}M$$ be obtained by contracting the point process M by $$\epsilon$$ ; so, if m is a point of M, $$\epsilon$$ m is a point of $$S_{\epsilon}M$$. Now if a kth generation person is at x, his children’s positions, relative to his own, are given by an independent copy of $$S_{\epsilon}M_{k\epsilon,x}.$$
Let $$P^{\epsilon}_{k,x}$$ be the law of the process resulting from a single kth generation person at x. If $$\zeta_ n$$ is the point process of the nth generation people then the main objective of this paper is to establish lower bounds on $\liminf \epsilon \log (P^{\epsilon}_{0,x}(\zeta_ n[y-u,y+u]>0),$ as $$\epsilon\to 0$$, where $$n=[t/\epsilon]$$ and $$u=\sigma \sqrt{\epsilon}$$. Obviously interest centres on y values for which the probability here is small (and these, in a sense, constitute the subcritical region of the process). Techniques from branching processes and large deviation theory are used; the lower bounds obtained arise from ’action integrals’ defined through the Cramér transform of the intensity measure of $$M_{t,x}$$.
Reviewer: J.D.Biggins

##### MSC:
 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60F10 Large deviations 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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