Supercritical superprocesses: Proper normalization and non-degenerate strong limit
Suppose that X = { X t , t ⩾ 0;ℙ μ } is a supercritical superprocess in a locally compact separable metric space E . Let φ 0 be a positive eigenfunction corresponding to the first eigenvalue λ 0 of the generator of the mean semigroup of X . Then M t := e − λ 0 t 〈 ϕ 0 , X t 〉 is a positive martingal...
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Published in | Science China. Mathematics Vol. 62; no. 8; pp. 1519 - 1552 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Beijing
Science China Press
01.08.2019
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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Summary: | Suppose that
X
= {
X
t
,
t
⩾ 0;ℙ
μ
} is a supercritical superprocess in a locally compact separable metric space
E
. Let
φ
0
be a positive eigenfunction corresponding to the first eigenvalue λ
0
of the generator of the mean semigroup of
X
. Then
M
t
:=
e
−
λ
0
t
〈
ϕ
0
,
X
t
〉
is a positive martingale. Let
M
∞
be the limit of
M
t
. It is known (see Liu et al. (2009)) that
M
∞
is non-degenerate if and only if the
L
log
L
condition is satisfied. In this paper we are mainly interested in the case when the
L
log
L
condition is not satisfied. We prove that, under some conditions, there exist a positive function γ
t
on [0, ∞) and a non-degenerate random variable
W
such that for any finite nonzero Borel measure
μ
on
E
,
M
t
:=
e
-
λ
0
t
〈
φ
0
,
X
t
〉
We also give the almost sure limit of γ
t
〈
f, X
t
〉 for a class of general test functions
f
. |
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ISSN: | 1674-7283 1869-1862 |
DOI: | 10.1007/s11425-018-9402-4 |