Kritische Verzweigungsprozesse mit allgemeinem Phasenraum. IV

Liemant, Alfred

doi:10.1002/mana.19811020120

Cited by 10 publications

(4 citation statements)

References 4 publications

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“…A multitype generalization can be found in [PR77]. A characterization in terms of parameters in the model (as in Corollary 6) was provided in [DF85] (even in a random environment context), using a backward tree technique developed in [Kal77] and [Lie81]. See [DFFP86] for such a parametrized condition in a superprocess setting, and [GW91] in a continuous time particle version.…”

Section: Resultsmentioning

confidence: 99%

An integral test for a critical multitype spatially homogeneous branching particle process and a related reaction-diffusion system

Fleischmann

Vatutin

2000

Probab Theory Relat Fields

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An integral test (Theorem 5) is established for the dichotomy concerning local extinction and survival (even persistence) at late times for critical multitype spatially homogeneous branching particle systems in continuous time. Our conditions on the branching mechanism are close to the ones known from "classical" processes without motion component. This generalizes and complements results of López-Mimbela and Wakolbinger [LMW96] and others. Our approach is based on some genealogical tree analysis combined with the study of the long-term behavior of L 1 -norms of solutions of related systems of reaction-"diffusion" equations, which is perhaps also of some independent interest.

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Section: Resultsmentioning

confidence: 99%

An integral test for a critical multitype spatially homogeneous branching particle process and a related reaction-diffusion system

Fleischmann

Vatutin

2000

Probab Theory Relat Fields

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“…Obviously, the sequence of measures η (L n ) n is a.s. nondecreasing. In addition, the calculations given above (10) show that for all n ∈ N 0 the law of η…”

Section: Backward Step: ηmentioning

confidence: 92%

“…The construction of the infinite Palm tree in (the proof of) Lemma 3 is similar to the 'method of backward trees' in Kallenberg (1977) and Liemant (1981). However, in contrast to this earlier work, we are not only interested in a Palm version of the particles of generation n (i.e., of χ n ) but of all generations up to generation n (i.e., of χ n = n k=0 χ k ).…”

Section: Modelmentioning

confidence: 99%

“…, n}. Given the sequence (L n ), we can then construct a nondecreasing sequence of random measures η (10)), where in step n we make either a forward or a backward step-depending on the realized step size of L n . So what remains to be proved is that we can construct the forward step and the backward step in such a way that in both cases we only add new points and do not remove old points.…”

Section: Lemma 3 (Forward/backward Construction Of Infinite Palm Tree)mentioning

confidence: 99%

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Critical cluster cascades

Kirchner

2022

Adv. Appl. Probab.

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We consider a sequence of Poisson cluster point processes on $\mathbb{R}^d$ : at step $n\in\mathbb{N}_0$ of the construction, the cluster centers have intensity $c/(n+1)$ for some $c>0$ , and each cluster consists of the particles of a branching random walk up to generation n—generated by a point process with mean 1. We show that this ‘critical cluster cascade’ converges weakly, and that either the limit point process equals the void process (extinction), or it has the same intensity c as the critical cluster cascade (persistence). We obtain persistence if and only if the Palm version of the outgrown critical branching random walk is locally almost surely finite. This result allows us to give numerous examples for persistent critical cluster cascades.

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Kritische Verzweigungsprozesse mit allgemeinem Phasenraum, V

Liemant¹,

Matthes²

1982

Math. Nachr.

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EinleitungIn der vorliegenden Arheit werden in Gestalt von Theorem 15.1. liinreichende Redingungen an Y abgeleitet , unter denen jedes bezuglich x schauerinvariantegenugen. Offenbar ist H ein abgeschlossener linearer Unterraum von L f ( v ) . Fiir alle f aus H ergibt sich d. h. also Somit gilt (*) J f W W a ) = O VEH) * 3. Wir wollen nun zusatziich annehmen, v gehore zu X p ni'?fP". Wegen VCN;~" erhalten wir fur alle X aus 23 _ -~ lim \l(J)* ( I = --1 , * J)\\lim Illx * P I -I, * J [~+ ' ] /~~= O . f l --l -r k , -Genugt ein g aus L _ ( v ) der Bedingung l,(h): = J~( u ) g(a) v(du) = 0 (hE H ) , so folgt wegen (**) fiir alle f aus LI(v) W=J (4f)) ( a ) s(a) v(da) =J ( f + : # J ) ( a ) g(a) v(da) -$ ( f -Y J ) ( a ) g(n) v(da) . Liemant,/Matthes, Kritisrhe Verzweigungsprozesse 237 Setzen wir nun y : = J g(a) 9 0 so existiert ein 1 1 0 mit der Eigenschaft und ivir erhaiten O~~: = l v + y~2 2 1 v , J f ( a ) o ( d a ) = J f + ( a ) ( J * 0) ( d a ) -J f -( a ) ( J * 0) ( W = $ f ( a ) ( J * 0) @a). d. h. 0 liegt in N J . Wegen v c N p ist 0 und damit auch das signierte Ma13 y ein Vielfaches von v. Somit ist lg ein Vielfaches von Ii. Zusammenfassend konnen wir feststellen : Jede stetige Linearform auf L,(v), die auf dem abgeschlossenen linearen Unterraum H identisch verschwindet, ist ein Vielfaches von ll. In Anbetracht von (*) fallt daher H mit (h : hELI(v), li(h)= 0) zusammen. Es seien nun fl, f z irgendwelche nichtnegative %-mefibare reelle Funktionen auf A mit der Eigenschaft Mit Hilfe der oben abgeleiteten Charakterisierung von H konnen wir auf ffi) E H schliel3en. Nach 7.8. haben wir damit bewiesen, da13 (#"I) eine v-regulare Folge hildet. w $ f . ( a ) v ( d a ) =~ ( i = I , 2 ) . Als Gegenstiick zu Satz 9.1. ergibt sich (vgl. Abschnitt 4 in [I]).

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Kritische Verzweigungsprozesse mit allgemeinem Phasenraum. IV

Cited by 10 publications

References 4 publications

An integral test for a critical multitype spatially homogeneous branching particle process and a related reaction-diffusion system

An integral test for a critical multitype spatially homogeneous branching particle process and a related reaction-diffusion system

Critical cluster cascades

Kritische Verzweigungsprozesse mit allgemeinem Phasenraum, V

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