A zero-one law of almost sure local extinction
for (1 + β)-super-Brownian motion
Xiaowen Zhou 1
Department of Mathematics and Statistics, Concordia University,
1455 de Maisonneuve Blvd. West, Montreal, Quebec, H3G 1M8, Canada
E-mail address:[email protected]
Abstract
This paper considers the almost sure local extinction for d-dimensional (1 + β)-superBrownian motion X starting at Lebesgue measure on Rd . Given closed balls (Bg(t) )t>0 centered
at 0 and with radiuses g(t) that are non-decreasing and right continuous in time t, let
τ := sup{t : Xt (Bg(t) ) > 0}.
For dβ < 2, it is shown that P{τ = ∞} is equal to either 0 or 1 depending on whether the
R∞
value of integral 1 g(y)d y −1−1/β dy is finite or infinite. An asymptotic bound for P{τ > t} is
given when P{τ < ∞} = 1.
Keywords: (1+β)-super-Brownian motion, almost sure local extinction, zero-one law, historical
super-Brownian motion, integral test
AMS Subject Classification 2000: 60G57, 60J80
1. Introduction
Given a positive integer d and a constant d < p < d + 2, write φp (x) := (1 + |x|2 )−p/2 , x ∈
Rd ; write Fp for the collection of all nonnegative continuous functions φ on Rd such that
supx φ(x)/φp (x) < ∞; write Mp for the set of all measures µ on Rd such that µ(φp ) < ∞.
For 0 < β ≤ 1 the d-dimensional (1 + β)-super-Brownian motion X with branching rate
γ > 0 and initial measure m ∈ Mp is a measure-valued stochastic process with its probability
law specified by the Laplace functional
h
i
Pm e−Xt (φ) = e−m(Vt φ) ,
where for φ ∈ Fp , v(t, x) := Vt φ(x) solves the nonlinear p.d.e.
∂v(t, x)
1
γ
= ∆v(t, x) −
v(t, x)1+β , v(0, x) = φ(x).
∂t
2
1+β
1This work is supported by an NSERC grant.
1
2
Such a process arises from the scaling limit of empirical measures of critical branching motions in which the offspring distributions are in the domain of attraction for stable law.
The persistence-extinction dichotomy is well known for super-Brownian motion. In low dimensions the super-Brownian motion X, started at Lebesgue measure λ, suffers local extinction;
i.e. Xt (A) → 0 in probability as t → ∞ for any compact set A. On the other hand, in high
dimensions X started at λ is persistent; i.e. the limiting random measure X∞ of (Xt ) satisfies
P[X∞ ] = λ, and consequently the local extinction can not happen. It is known that X is
persistent if and only if dβ > 2. For the case dβ = 2, Xt (A) converges to 0 in probability,
but not almost surely. See [1] for the earliest work and Sections 5.2 and 5.3 of [7] for a more
detailed discussion.
It was first proved in [10] that almost surely Xt (A) = 0 for t large enough if X is the onedimensional 2-super-Brownian motion, and we say X suffers almost sure local extinction. A key
fact which was first obtained in [9] was that the Laplace functional of weighted local time for
super-Brownian motion can be expressed in terms of nonlinear p.d.e. The proof for the almost
sure local extinction then relied on analyzing the weighted local time using the corresponding
p.d.e..
A stronger version of the local extinction property can be found in Proposition 3.2 of [8]. For
any η, it is shown that starting with Lebesgue measure λ on R, almost surely Xt ([−tη , tη ]) = 0
for t large enough. Also see Proposition 7 of [4] for another result along this line. We also
refer to [8] and [4] for the applications of such results in the study of super-Brownian motion
in random medium.
In this paper we further improve the known results on almost sure local extinction for (1+β)super-Brownian motion with Lebesgue initial measure on Rd . For d < 2/β, given (Bg(t) ), a
collection of closed balls in Rd with non-decreasing and right continuous radiuses g(t), we show
that almost surely Xt (Bg(t) ) = 0 for t large enough provided
Z
∞
g(y)d y −1−1/β dy < ∞.
1
Our approach is similar to those in [8] and [10]. It exploits the additivity (branching property), the extinction probability for super-Brownian motion and the asymptotic speed of propagation for super-Brownian motion support. To illustrate the idea we only consider the case
such that d = 1, β = 1 and At = [−ta , ta ], 0 < a < 1, the proof can be sketched as follows.
For large t the super-Brownian motion started at the Lebesgue measure restricted on interval
[−ta − tδ , ta + tδ ] with 1/2 < δ < 1 can only survive up to time t with probability of the order
ta∨δ−1 . In addition, for large t the “front” of support for the super-Brownian motion started at
the Lebesgue measure restricted on the complement of the interval [−tα − tδ , tα + tδ ] can not
move toward 0 at a speed much faster than t1/2 . Consequently, the support for the later pro0
cess can only intersect interval [−tα , tα ] with probability of the order e−tδ for 0 < δ 0 < 2δ + 1.
3
Choosing a sequence (tn ) that increases geometrically, the desired result then follows readily
form Borel-Cantelli lemma.
On the other hand, with d < 2/β, for any constant c > 0 the time-space-mass scaling
−1/β
t
Xt (Bct1/dβ ) converges in distribution to a Poisson sum of i.i.d. Gamma random variables
as t → ∞; see Theorem 3.1 of [3]. It follows that
lim sup P{Xt (Bct1/dβ ) > 0} > 0.
t→∞
Consequently, the above-mentioned almost sure local extinction can not happen for (Bct1/dβ ).
In this paper we also find a more general condition in terms of a similar integral test on g, under
which the almost sure local extinction does not occur for X with respect to regions (Bg(t) ).
More precisely, we show that
P{sup{t : Xt (Bg(t) ) > 0} = ∞} = 1
if
Z
∞
g(y)d y −1−1/β dy = ∞.
1
For d = 1, β = 1 and g(t) = 2t/ ln t for t large enough, the proof proceeds as follows. For
tn := 2n , n = 1, 2, . . ., write X n for independent super-Brownian motion with initial measure the
Lebesgue measure restricted to interval (tn−1 / ln tn−1 , tn / ln tn ), respectively. The extinction
probability for super-Brownian motion gives P{Xtnn (1) > 0} ≥ c/n for some c > 0 and for all n
large enough. It then follows from Borel-Cantelli lemma that with probability one Xtnn (1) > 0
for infinitely many n. Since the support for X n can not propagate much faster than t1/2 , by
Borel-Cantelli again Xtnn (R − [−g(tn ), g(tn )]) > 0 for only finitely many n. The desired result
follows.
These integral tests on almost sure local extinction for super Brownian motion answer a
question asked by Edwin Perkins.
This paper is arranged as follows. Using the modulus of continuity for historical superBrownian motion, in Section 2 we first present several estimates on the hitting probability of
Br by the super-Brownian motion. We then obtain the zero-one law on the almost sure local
extinction in Section 3. An old result is also revisited in Section 3.
2. Preliminaries
Recall that for m ∈ Mp the Laplace functional for the total mass has the expression
( µ
)
¶1
β
1+β
(2.1)
Pm [exp{−αXt (1)}] = exp −
αm(1) ,
1 + β + γβtαβ
which implies the extinction probability
(2.2)
)
( µ
¶1
1+β β
m(1) .
Pm {Xt (1) = 0} = exp −
γβt
4
Write MFt (D) for the set of finite measures on D(Rd ) supported by paths which are constant
after time t. Let H, under Q0,m , be the historical (1+β)-super-Brownian motion corresponding
to X. Then Ht is MFt (D)-valued with Xt as its projection at t. It keeps track of the history
of all the “individuals” in a super-Brownian motion that are still “alive” at time t . We refer
to [6] and [11] for explicit definition and more discussions on historical processes. To prove
Lemma 2.2 we need a result on the historical modulus of continuity for H. Such a result is well
known for 2-super-Brownian motion. For (1 + β)-super-Brownian motion it was proved in [5].
Lemma 2.2 can be obtained by following the line of proof for Theorem III.1.3. of [11]. Write
S(Ht ) for the support for Ht .
Lemma 2.1. Given a d-dimensional historical (1 + β)-super-Brownian motion H and m ∈
MF0 (D), for c > 2 large enough there exists a random variable ∆ > 0 such that Q0,m a.s. for
all t ≥ 0,
S(Ht ) ⊂ {y ∈ C(R) : |y(r) − y(s)| ≤ c|(r − s) log(|r − s|)|1/2 , ∀r, s > 0, |r − s| ≤ ∆}.
Moreover, there are constants ρ = ρ(β) > 0 and c = c(d, β, c) > 0 such that
Q0,m {∆ ≤ r} ≤ cm(1)rρ .
(2.3)
For any A ∈ B(Rr ) write λ1(A) for the Lebesgue measure restricted on A, i.e. λ1(A)(B) =
λ(A ∩ B). Write Br := {x ∈ Rd : |x| ≤ r} for r > 0. Write Xr = Xr,· for a d-dimensional
(1 + β)-super-Brownian motion with initial measure Xr,0 = λ1(Bcr ), Bcr = Rd − Br . Write Sr,·
for the support process for Xr,· .
The following result gives an estimate on the speed of propagation for a super-Brownain
motion over time. Roughly, its support can not move much faster that t1/2 .
Lemma 2.2. Given a > 0 and δ > 1/2, for any δ 0 with 0 < δ 0 < 2δ − 1 we have
(2.4)
©
ª
0
lim exp{tδ } sup P ∃s ≤ t, Xr+tδ ,s (Br ) > 0 = 0.
t→∞
r≤ta
Proof. Given t > 0 and 0 < r < ta , write X·j for independent historical super-Brownian motion
with initial measures
mj := λ1(Br+tδ +tδ(j+1) − Br+tδ +tδj ), j = 0, 1, 2, . . . .
Write H·j for the corresponding historical super-Brownian motion. Note that we have suppressed the dependence of X·j and H·j on both t and r. Put
A(t0 , r0 ) := {y ∈ C(Rd ) : inf0 |y(s)| ≤ r0 }.
s≤t
5
l
m
l
m
00
00
Given δ 0 < δ 00 < 2δ − 1, let l0 := t exp{tδ } and lj := t exp{tδ j } for j > 0. Then for t
large enough we have
∞
©
ª X
©
ª
P ∃s ≤ t, Xr+tδ ,s (Br ) > 0 ≤
P ∃s ≤ t, Xsj (Br ) > 0
j=0
(2.5)
≤
∞
X
©
ª
Q0,mj ∃s ≤ t, Hsj (A(s, r)) > 0 .
j=0
n
o
To estimate Q0,mj ∃s ≤ t, Hsj (A(s, r)) > 0 , by Lemma III.1.2. (for (1+β)-branching superBrownian motion) in [11], for j > 0 we first have
n
o
j
Q0,mj H(i+1)t/l
(A(it/l
,
r
+
1))
>
0
j
j
"
( µ
)#
¶1
1+β β j
= 1 − Q0,mj exp −
Hit/lj (A(it/lj , r + 1))
γβt/lj
µ
¶1
i
h
1
1+β β
j
β
(A(it/l
,
r
+
1))
.
≤ lj
Q0,mj Hit/l
j
j
γβt
Since the mean measure of Hj (r) under Q0,mj is the Wiener measure stopped at time r, then
for 1 ≤ i ≤ lj − 1 we then have
i
h
j
(A(it/l
,
r
+
1))
≤ mj (1)P
Q0,mj Hit/l
j
j
≤
(
2dmj (1)
√
2π
)
sup |Ws | > tδ + tδj − 1
s≤it/lj
Z
∞
tδ +tδj −1
idt/lj
√
e−x
2 /2
dx
½ 2δj ¾
t
≤ mj (1) exp −
,
2dt
where W is a d-dimensional Brownian motion starting at 0 and t is large enough.
Let
Tj := inf{t : X j (Br ) > 0}
with the convention inf ∅ = ∞. For Tj < ∞ choose a nonnegative integer i such that it/lj ≤
Tj < (i + 1)t/lj . Then for i ≥ 1, by time (i − 1)t/lj either the super-Brownian motion X j has
already charged the set Br+1 or it has not charged Br+1 yet. In the later case we must have
∆ ≤ 2t/lj because the support process for X j has to travel a distance of at least 1 between
time (i − 1)t/lj and time T . For i = 0 the support process also has to travel a long distance
between time 0 and time T , which implies ∆ < t/lj . Putting these together, by Lemma 2.1 we
6
have
©
ª
Q0,mj ∃s ≤ t, Hsj (A(s, r)) > 0
lj −1
≤
X
i=1
o
n
j
(A(it/l
,
r
+
1))
>
0
+ Q0,mj {∆ ≤ 2t/lj }
Q0,mj H(i+1)t/l
j
j
¶1
½ 2δj ¾
1 + β β β1
t
00
≤
lj mj (1) exp −
+ cmj (1)2ρ exp{−ρtδ j }
γβt
2dt
i=1
¶1
µ
½ 2δj−1 ¾
1 + β β 1+ β1
t
00
lj mj (1) exp −
+ cmj (1)2ρ exp{−ρtδ j }.
≤
γβt
2d
P∞
Since X has the same distribution as j=0 Xj , limit (2.4) follows from (2.5) and (2.6).
(2.6)
lj −1 µ
X
¤
We can make δ depend on t in Lemma 2.2 by modifying the proof for Lemma 2.2.
Lemma 2.3. Given a > 0 and k > 0, for the processes considered in Lemma 2.2 there exists
k 0 > 0 such that
n
o
lim tk sup P ∃s ≤ t, Xr+k0 (t ln t)1/2 ,s (Br ) > 0 = 0.
t→∞
r≤ta
Proof. We can choose 1/2 < δ < 1. Choose q satisfying qρ − d(a ∨ δ) > k and put lj =
dt1+qj e, j ≥ 0. Put
mj = λ1(Br+k0 (t ln t)1/2 +tδ(j+1) − Br+k0 (t ln t)1/2 +tδj ), j ≥ 0,
where
k 02 > 2d [d(a ∨ δ) + (1 + q)(1 + 1/β) + k] .
Then just follow the line of proof for Lemma 2.2.
¤
Remark 2.4. In dimension one both Lemma 2.2 and Lemma 2.4 can be improved in the sense
that the supreme is taken for all t > 0 instead of for 0 < t ≤ ta .
Remark 2.5. If the Xr in Lemma 2.2 is replaced by a d-dimensional (1 + β)-super-Brownian
motion with initial measure Xr,0 = λ1(Bg(r) ), where g is any nonnegative increasing function
such that ∃δ > 0,
lim inf g(t)/t1/2+δ > 0,
t→∞
then by a similar argument we can show that for any δ 0 , δ 00 > 0,
n
³
´
o
0
lim g(t)δ P Xt,t Bc(1+δ00 )g(t) > 0 = 0.
t→∞
The proof for the next result also follows that for Lemma 2.2.
7
Lemma 2.6. Let (Xr,· ) be a collection of d-dimensional (1 + β)-super-Brownian motion with
initial measures Xr,0 = λ1(Br ), r > 0. Then for any a > 0, δ > 1/2 and 0 < δ 0 < 2δ − 1, we
have
ª
©
0
lim exp{tδ } sup P ∃s ≤ t, Xr,s (Bcr+tδ ) > 0 = 0.
t→∞
r≤ta
3. Main results
Let g(t), t ≥ 0, be any nonnegative, nondecreasing and right continuous function. Define
τ := sup{t ≥ 0 : Xt (Bg(t) ) > 0}.
τ is then the first time when almost sure local extinction occurs with respect to (Bg(t) ). It is
not a stopping time.
Theorem 3.1. Let X be a d-dimensional (1 + β)-super-Brownian motion such that X0 = λ
and dβ < 2. If
Z ∞
−1− β1
dy < ∞,
(3.1)
g(y)d y
1
then
(3.2)
P{τ < ∞} = 1.
Moreover,
(3.3)
µ
Z
d
d
− β1
2
(ln t) 2 ∨
lim sup P{τ > t} t
t→∞
∞
1
d −1− β
g(y) y
¶−1
dy
< ∞.
t
Proof. For n = 1, 2, . . . put
tn = en and rn = g(tn+1 ) + k 0 (tn+1 ln tn+1 )1/2 ,
where k 0 is obtained from Lemma 2.3 for a = (1 + β −1 )/d and k = 2. Throughout the proof for
part (i) we write X·n and Y·n for independent (1+β)-super-Brownian motion with X0n = λ1(Brn )
and Y0n = λ1(Bcrn ), respectively. Then X· has the same distribution as X·n + Y·n .
Condition (3.1) and the monotonicity of g give
lim g(y)d y
y→∞
−1− β1
= 0.
Then it follows form Lemma 2.3 that for m large enough and n ≥ m,
P{∃t < tn+1 , Ytn (Bg(tn+1 ) ) > 0} ≤ t−2
n+1 .
8
Applying (2.2) we have
P{∃t ≥ tn , Xtn (Bg(tn+1 ) ) > 0}
≤ P{∃t ≥ tn , Xtn (Rd ) > 0}
à µ
!
¶1
1+β β
= 1 − exp −
λ(Brn )
γβtn
¶1
µ
1 + β β d d − β1
≤
2 rn tn
γβ
µ
¶1
´ −1
d
1+β β d ³
≤
2 d g(tn+1 )d + k 0d (tn+1 ln tn+1 ) 2 tn β
γβ
µ
¶1
µ
¶
1
d
1
d
1 + β β d β1
d −β
0d 2 − β
≤
2 de
g(tn+1 ) tn+1 + k tn+1 (ln tn+1 ) 2 .
γβ
Plainly,
P{∃tn ≤ t < tn+1 , Xt (Bg(tn+1 ) ) > 0}
≤ P{∃tn ≤ t < tn+1 , Xtn (Bg(tn+1 ) ) > 0} + P{∃tn ≤ t < tn+1 , Ytn (Bg(tn+1 ) ) > 0}.
Moreover, for m large enough,
∞
X
P{∃tn ≤ t < tn+1 , Xtn (Bg(tn+1 ) ) > 0}
n=m
(3.4)
¶1
µ
¶
∞ µ
1
d
1
X
d
1 + β β d β1
d −β
0d 2 − β
2
g(tn+1 ) tn+1 + k tn+1 (ln tn+1 )
≤
2 de
γβ
n=m
µ
¶ 1 µZ ∞
¶
Z ∞
1
d
1
1
1+β β
0d
x 2−β
x d2
d β
x d x−1 − β
(e )
(ln e ) dx
≤ d2 e
g(e ) (e ) dx + k
γβ
m+1
m+1
!
¶1 Ã Z ∞
µ
Z ∞
β
1
1
1
1
d
d
1
+
β
− −1
−1− β
eβ
≤ d2d e β
dy + k 0d
g(y)d y
y 2 β (ln y) 2 dy
γβ
tm+1
tm+1
!
1 ÃZ
µ
¶
Z ∞
³ 1
´ 1+β β
∞
1
1
d
1
d
−1−
−
−1
0d
d
d β
β dy ∨
g(y) y
y 2 β (ln y) 2 dy .
≤ d2 e e β + k
γβ
tm+1
tm+1
Then Borel-Cantelli lemma together with the assumptions of Theorem 3.1 implies that almost
surely, for n large enough, Xt (Bg(tn+1 ) ) = 0 for all t > tn .
Finally, observe that
{(x, t) : tn < t ≤ tn+1 , x ∈ Bg(t) } ⊂ {(x, t) : t > tn , x ∈ Bg(tn+1 ) }.
The desired result (3.2) follows from
{(x, t) : t > tm , x ∈ Bg(t) } ⊂
[
{(x, t) : tn < t ≤ tn+1 , x ∈ Bg(tn+1 ) }.
n≥m
9
The estimate (3.3) follows form (3.4) and the inequality
Z ∞
d
d
d
d
4β
− 1 −1
−1
y 2 β (ln y) 2 dx <
t 2 β (ln t) 2 .
2 − dβ
t
.
¤
Remark 3.2. The estimate (3.3) reveals that E(τ ) < ∞ for g(t) = tα with α ≥ 0 if
(1/2 ∨ α)d < 1/β − 1.
To justify the sharpness of Theorem 3.1 we present the next result. The zero-one law follows
readily.
Theorem 3.3. For dβ < 2 let X be a d-dimensional (1 + β)-super-Brownian motion such that
X0 = λ. If
Z ∞
−1− β1
(3.5)
g(y)d y
dy = ∞,
1
then
(3.6)
P{τ = ∞} = 1.
Proof. We can and will assume that g(1)/3 < 1. Choose ² > 0 satisfying
(3.7)
d/2 − 1/β + d² < 0.
Because of (3.5) and (3.7) either g(t)/3 ≥ t1/2+² for all t large enough or there exists a finite,
strictly increasing sequence (si ) such that s1 := 1,
s2i := inf{s > s2i−1 : g(s)/3 ≥ 2s1/2+² }
and
s2i+1 := inf{s ≥ s2i : g(s)/3 < s1/2+² }
for i = 1, 2, . . ..
We only prove (3.6) for the second case, which is more involved. In this case observe that
si → ∞,
(3.8)
g(s2i −)/3 ≤ 2s1/2+² ≤ g(s2i )/3
and
g(s2i−1 −)/3 = g(s2i−1 )/3 = s1/2+² ,
where g(s−) denotes the left limit of g(·) at s. In addition,
(3.9)
g(s)/3 ≤ 2s1/2+² for s ∈ (s2i−1 , s2i )
and
g(s)/3 ≥ s1/2+² for s ∈ (s2i , s2i+1 ).
10
We further define strictly increasing sequences (ti ) and (ni ) such that tni = si ; tk ≤ 2tk−1 for
k = n2i−1 + 1, . . . , n2i ; and
tk := inf{tk−1 < t ≤ s2i+1 : g(t) ≥ 2g(tk−1 )}
for k = n2i + 1, . . . , n2i+1 , where we use the convention that inf ∅ := s2i+1 . Notice that
2g(tn2i ) < g(tn2i+2 ) and
(3.10)
g(tk+1 ) ≥ 2g(tk ) ≥ g(tk+1 −) for n2i ≤ k ≤ n2i+1 − 1.
Put rn := g(tn+1 −)/3. Write (X·n )n≥1 for independent (1 + β)-super Brownian motions
¢
¡
starting at λ1(Brn − Brn−1 ) n≥1 . To reach (3.6) it suffices to show that with probability one,
Xtjj (Bg(tj ) ) > 0 for infinitely many j.
Observe that
Z tn
Z tn
−1− β1
−1− β1
dy ≤ g(tn −)d
dy
g(y)d y
y
t
t
n−1
n−1
(3.11)
−1
−1
β
≤ βg(tn −)d (tn−1
− tn β ).
In addition,
(3.12)
( µ
)
¶1
β
1
+
β
P{Xtnn (1) > 0} = 1 − exp −
λ(Brn − Brn−1 )
γβtn
( µ
)
¶1
1 1+β β
−1
≥ min
λ(Brn − Brn−1 ), 1 − e
3 γβtn
)
( µ
¶1
³
´
1 1+β β
c(d) g(tn+1 −)d − g(tn −)d , 1 − e−1
≥ min
3 γβtn
where c(d) is a positive constant and we have used inequality 1 − e−x ≥ x/3 for 0 ≤ x ≤ 1.
Since
∞
∞
³
´
X
X
−1
−1
− β1
−1
tn β g(tn+1 −)d − g(tn −)d ≥ −t1 β g(t1 −)d +
g(tn −)d (tn−1
− tn β ),
n=1
n=2
combining (3.5), (3.11) and (3.12) we have
∞
X
(3.13)
P{Xtnn (1) > 0} = ∞.
n=1
Further, we define a sequence (aj ) such that

g(t −) for n
j
2i−1 ≤ j ≤ n2i , i = 1, 2, . . . ;
aj :=
g(tn −) for n2i < j < n2i+1 , i = 1, 2, . . . .
2i
11
Then
∞ nX
2i −1
X
P{Xtjj (1) > 0} ≤
³
´
−1
tj β c(d, β, γ) g(tj+1 −)d − g(tj −)d
∞ nX
2i −1
X
i=1 j=n2i−1
i=1 j=n2i−1
≤ c(d, β, γ)
∞
X
− β1
tj
³
´
adj+1 − adj
j=1
≤ c(d, β, γ)
(3.14)
∞
X
− β1
adj+1 (tj
−1
β
− tj+1
)
j=1
≤ c(d, β, γ)
∞
1
X
− β1
1/2+² d − β
)
(12tj
) (tj − tj+1
j=1
Z
≤ c∗ (d, β, γ)
∞
td/2−1/β+d²−1 dt
t1
< ∞,
where c(d, β, γ) and c∗ (d, β, γ) are the obvious constants. To obtain the third inequality of
(3.14) we use the fact that by (3.7) and (3.9),
−1/β
tj
g(tj+1 −)d → 0 for n2i−1 ≤ j ≤ n2i − 1 and i → ∞.
We also need (3.8) to obtain the fourth inequality.
Inequalities (3.13) and (3.14)together leads to
∞ n2i+1
X
X−1
P{Xtjj (1) > 0} = ∞.
i=1 j=n2i
It then follows readily from Borel-Cantelli lemma that with probability one, Xtjj (1) > 0 for
infinitely many j with n2i ≤ j < n2i+1 and i = 1, 2, . . ..
Therefore, to finish the proof we only need to show that with probability one, Xtjj (Bcg(tj ) ) = 0
for any n2i ≤ j < n2i+1 and i large enough. To this end, we might use Remark 2.5 together
with (3.10) and the fact that for n2i ≤ j < n2i+1 ,
1
g(tj ) − rj = g(tj ) − g(tj+1 −)
3
2
≥ g(tj ) − g(tj )
3
1 1/2+²
,
≥ tj
3
and again, the Borel-Cantelli lemma.
¤
Remark 3.4. In the case dβ < 2, given ² > 0 we have P{τ < ∞} = 1 when
g(t) = t1/dβ (ln t)−1/d (ln ln t)−(1+²)/d
12
for large t. On the other hand, P{τ = ∞} = 1 when
g(t) = t1/dβ (ln t)−1/d (ln ln t)−1/d
for large t.
Remark 3.5. Notice that condition (3.5) can not guarantee the existence of a finite random
time T such that Xt (Bg(t) ) > 0 for all t > T . For instance, given d = β = 1, using (2.2) and
Lemma 2.2 one can show that
lim inf P{Xt (Bt ) = 0} > 0.
t→∞
Consequently such a T does not exist for g(t) = t.
Remark 3.6. For t > 0 put
It := inf{r ≥ 0 : Xt (Br ) > 0}.
For any g satisfying (3.1) and any c > 0, by Theorem 3.1 we have almost surely,
lim inf It /cg(t) ≥ 1.
t→∞
Therefore,
lim inf It /g(t) = ∞.
t→∞
Similarly, for any g satisfying (3.5), we have almost surely,
lim inf It /g(t) = 0.
t→∞
Hence, we would not expect law of iterated logarithm type results for It as t → ∞.
At the end of this paper we want to point out that the result on the time-space scaling limit
of X mentioned in the introduction can also be recovered using Lemma 2.2.
Proposition 3.7. (Dawson and Fleischmann) Given c > 0, for the process X in Theorem 3.1
we have
( µ
)
¶1
oi
h
n
β
1
+
β
(3.15)
lim P exp −αt−1/β Xt (Bct1/dβ ) = exp −
αcd vd ,
t→∞
1 + β + γβαβ
where vd :=
2π d/2
dΓ(d/2)
denotes the volume of the d-dimensional sphere with radius 1. Therefore,
the scaled process Zt := t−1/β Xt (Bct1/dβ ) has a limiting distribution with its Laplace transform
determined by (3.15).
Proof. Choose 1/2 < δ < 1/dβ. In this proof for each t > 0 we write Xt,· , Yt,· and Zt,·
for independent (1 + β)-super-Brownian motions with Xt,0 = λ1(Bct1/dβ −tδ ), Yt,0 = λ1(Rd −
Bct1/dβ +tδ ) and Zt,0 = λ1(Bct1/dβ +tδ − Bct1/dβ −tδ ), respectively.
13
Clearly, Xt,t (Rd − Bct1/dβ ) → 0 in probability by Lemma 2.6, and Yt,t (Bct1/dβ ) → 0 in probability by Lemma 2.2. In addition, t−1/β Zt,t (1) → 0 in probability. As a result, it follows from
(2.1) that
h
n
oi
lim P exp −αt−1/β Xt (Bct1/dβ )
t→∞
h
n
oi
= lim P exp −αt−1/β Xt,t (Bct1/dβ )
t→∞
h
n
oi
= lim P exp −αt−1/β Xt,t (1)
t→∞
( µ
)
¶1
β
1+β
= lim exp −
αt−1/β λ(Bct1/dβ −tδ )
t→∞
1 + β + γβαβ
( µ
)
¶1
β
1+β
d
= exp −
αc vd .
1 + β + γβαβ
¤
Remark 3.8. If we choose a different scaling, similar to the proof for Proposition 3.7, using
Lemma 2.2 we can show that
oi
n
o
h
n
lim P exp −λt−ξ Xt (Bctξ/d ) = exp −λcd vd
t→∞
for the process X in Proposition 3.1 and for ξ > 1/β. i.e. t−ξ Xt (Bctξ/d ) converges weakly to
the volume of Bc , which is well known.
On the other hand, for ξ < 1/β
oi
h
n
lim P exp −λt−ξ Xt (Bctξ/d ) = 1.
t→∞
Therefore, t−ξ Xt (Bctξ/d ) converges to 0 weakly as t → ∞.
References
[1] D. A. Dawson, The critical measure difusion, Z. Wahr. verw. Geb. 40, (1977) 125–145.
[2] D. A. Dawson and K. Fleischmann, Critical dimension for a model of branching in a radom medium, Z.
Wahr. verw. Geb. 70, (1985) 315–334.
[3] D. A. Dawson and K. Fleischmann, Strong clumping of critical space-time branching models in subcritical
dimensions, Stoch. Proc. Appl. 30 (1988) 193–208.
[4] D. A. Dawson and K. Fleischmann, A continuous super-Brownian motion in a super-Brownian medium, J.
Theor. Probab. 10 (1997) 213–276.
[5] D. A. Dawson, K. J. Hochberg and V. Vinogradov, High-density limits of hierarchically structured
branching-diffusion populations, Stoch. Proc. Appl. 62 (1996) 191–222.
[6] D. A. Dawson and E. A. Perkins, Historical processes, Mem. Amer. Math. Soc. 93 no. 454, 1991.
[7] D. A. Dawson and E. A. Perkins, Measure-valued processes and renormalization of branching particle
systems, in: R. A. Carmona and B. Rozovskii, eds., Stochastic partial differnetial equations: six perspectives,
Mathematical surveys and monographs No. 64, AMS, 1998.
14
[8] K. Fleischmann, A. Klenke and J. Xiong, Pathwise convergence of a rescaled super-Brownian catalyst
reactant process, J. Theor. Probab. 19 (2006) 557–588.
[9] I Iscoe, A weighted occupation time for a class of measured-valued branching processes, Probab. Th. Rel.
Fields 71 (1986) 85–116.
[10] I Iscoe, On the support of measure-valued critical branching Brownian motion, Ann. Probab. 16 (1988)
200–221.
[11] E. A. Perkins, Dawson-Watanabe superprocesses and measure-valued diffusions, Ecole d’été de probabilités
de Saint Flour, Springer-Verlag, 2000.