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Harris SC, Horton E, Kyprianou AE, Wang M. Yaglom limit for critical nonlocal branching Markov processes. ANN PROBAB 2022. [DOI: 10.1214/22-aop1585] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
| | | | | | - Minmin Wang
- School of Mathematical and Physical Sciences, University of Sussex
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Gonzalez I, Horton E, Kyprianou AE. Asymptotic moments of spatial branching processes. Probab Theory Relat Fields 2022. [DOI: 10.1007/s00440-022-01131-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
AbstractSuppose that $$X =(X_t, t\ge 0)$$
X
=
(
X
t
,
t
≥
0
)
is either a superprocess or a branching Markov process on a general space E, with non-local branching mechanism and probabilities $${\mathbb {P}}_{\delta _x}$$
P
δ
x
, when issued from a unit mass at $$x\in E$$
x
∈
E
. For a general setting in which the first moment semigroup of X displays a Perron–Frobenius type behaviour, we show that, for $$k\ge 2$$
k
≥
2
and any positive bounded measurable function f on E, $$\begin{aligned} \lim _{t\rightarrow \infty } g_k(t){\mathbb {E}}_{\delta _x}[\langle f, X_t\rangle ^k] = C_k(x, f), \end{aligned}$$
lim
t
→
∞
g
k
(
t
)
E
δ
x
[
⟨
f
,
X
t
⟩
k
]
=
C
k
(
x
,
f
)
,
where the constant $$C_k(x, f)$$
C
k
(
x
,
f
)
can be identified in terms of the principal right eigenfunction and left eigenmeasure and $$g_k(t)$$
g
k
(
t
)
is an appropriate deterministic normalisation, which can be identified explicitly as either polynomial in t or exponential in t, depending on whether X is a critical, supercritical or subcritical process. The method we employ is extremely robust and we are able to extract similarly precise results that additionally give us the moment growth with time of $$\int _0^t \langle f, X_t \rangle \mathrm{d}s$$
∫
0
t
⟨
f
,
X
t
⟩
d
s
, for bounded measurable f on E.
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