Digital Object Identifier (DOI) 10.1007/s00440-003-0316-9
Probab. Theory Relat. Fields 128, 606–628 (2004)
Renming Song
Sharp bounds on the density, Green function
and jumping function of subordinate killed BM
Received: 17 February 2003 / Revised version: 23 October 2003 /
Published online: 2 January 2004 –
c
Springer-Verlag 2004
Abstract. Subordination of a killed Brownian motion in a domain D ⊂ R
d
via an α/2-sta-
ble subordinator gives rise to a process Z
t
whose infinitesimal generator is −(−|
D
)
α/2
, the
fractional power of the negative Dirichlet Laplacian. In this paper we establish upper and
lower estimates for the density, Green function and jumping function of Z
t
when D is either
a bounded C
1,1
domain or an exterior C
1,1
domain. Our estimates are sharp in the sense that
the upper and lower estimates differ only by a multiplicative constant.
1. Introduction
Let X
t
be a d-dimensional Brownian motion in R
d
and T
t
an α/2-stable subordi-
nator starting at zero, 0 <α<2. It is well known that Y
t
= X
T
t
is a rotationally
invariant α-stable process whose generator is −(−)
α/2
, the fractional power of
the negative Laplacian. The potential theory corresponding to the process Y is the
Riesz potential theory of order α.
Suppose that D is a domain in R
d
, that is, an open connected subset of R
d
.
We can kill the process Y upon exiting D. The killed process Y
D
has been exten-
sively studied in recent years and various deep properties have been obtained. For
instance, when D is a bounded C
1,1
domain, sharp estimates on the Green function
of Y
D
were established in [4] and [14].
Let |
D
be the Dirichlet Laplacian in D. The fractional power −(−|
D
)
α/2
of the negative Dirichlet Laplacian is a very useful object in analysis and partial
differential equations, see, for instance, [19] and [16]. There is a Markov process
Z corresponding to −(−|
D
)
α/2
which can be obtained as follows: We first kill
the Brownian motion X at τ
D
, the first exit time of X from D, and then we subor-
dinate the killed Brownian motion using the α/2-stable subordinator T
t
. Note that
in comparison with Y
D
the order of killing and subordination has been reversed.
For the differences between the processes Y
D
and Z, see [18].
Despite its importance, the process Z has not been studied much. In [11], a rela-
tion between the harmonic functions of Z and the classical harmonic functions in D
R. Song: Department of Mathematics, University of Illinois, Urbana, IL 61801, USA.
e-mail: rsong@math.uiuc.edu
Mathematics Subject Classification (2000): Primary 60J45, Secondary 60J75, 31C25
Key words or phrases: Killed Brownian motions – Stable processes – Subordination – Frac-
tional Laplacian – Transition density – Green function – Jumping function
Subordinate killed Brownian motion 607
wasestablished. In [13] (see also [9]) the domain of the Dirichlet form of Z was iden-
tified when D is a bounded smooth domain and α = 1. In the recent paper [18] with
Vondracek, we studied the process Z in detail and established, among other things,
the behaviors of the jumping function J and the Green function G
D
of Z when D is
a bounded C
1,1
domain. Recall that a bounded domain D in R
d
, d ≥ 2, is said to be
a bounded C
1,1
domain if there exist positive constants r
0
and M with the following
property: for every z ∈ ∂D and r ∈ (0,r
0
], there exist a function
z
: R
d−1
→ R
satisfying the condition |∇
z
(ξ) −∇
z
(η)|≤M|ξ − η| for all ξ,η ∈ R
d−1
and
an orthonormal coordinate system CS
z
such that if y = (y
1
,...,y
d
) in the CS
z
coordinates, then
B(z, r) ∩ D = B(z, r) ∩{y : y
d
>
z
(y
1
,...,y
d−1
)}.
A bounded domain in R
1
is a finite open interval. So when speak of a bounded
C
1,1
domain in R
1
, we mean a finite open interval. It is well known that for a
bounded C
1,1
domain D, there exists r
1
> 0 depending only on D such that for
any z ∈ ∂D and r ∈ (0,r
1
], there exist two balls B
z
1
(r) and B
z
2
(r) of radius r such
that B
z
1
(r) ⊂ D, B
z
2
(r) ⊂ (D)
c
and {z}=∂B
z
1
(r) ∩ ∂B
z
2
(r). One of the main
results of [18] is the following
Theorem 1.1. Suppose that D is a bounded C
1,1
domain in R
d
and α ∈ (0, 2).
Let ρ(x) stand for the Euclidean distance between x and the boundary ∂D of D.
(1) There exist positive constants C
1
and C
2
such that for all x,y ∈ D,
C
1
ρ(x)ρ(y) ≤ J(x, y) ≤ C
2
ρ(x)ρ(y)
|x − y|
2
∧ 1
1
|x − y|
d+α
(2) If d>α, then there exist positive constants C
3
and C
4
such that for all
x,y ∈ D,
C
3
ρ(x)ρ(y) ≤ G
D
(x, y) ≤ C
4
ρ(x)ρ(y)
|x − y|
2
∧ 1
1
|x − y|
d−α
The lower bounds in the theorem above are very poor when |x − y| is small.
One of the main purposes of this paper is to establish sharp lower bounds which
differ from the upper bounds only by multiplicative constants. We are also going
to establish sharp estimates on the transition density of the process Z.
The content of this paper is organized as follows. In Section 2 we review some
preliminary results on subordinate killed Brownian motions obtained in [18]. In
Section 3 we review the sharp estimates obtained in [20] and [21] of the transition
density of killed Brownian in D when D is a bounded C
1,1
domain or an exterior
C
1,1
domain in R
d
(d ≥ 3) and extend the sharp estimates in the bounded domain
case to dimensions 1 and 2. In Section 4 we establish sharp estimates for the density,
Green function and the jumping function of Z when D is a bounded C
1,1
domain
or an exterior C
1,1
domain.
608 R. Song
2. Preliminary results on subordinate killed Brownian motion
Let X
1
= (
1
, F
1
, F
1
t
,X
1
t
,θ
1
t
, P
1
x
) be a d-dimensional Brownian motion in R
d
,
and let T
2
= (
2
, G
2
,T
2
t
, P
2
) be an α/2-stable subordinator starting at zero,
0 <α<2. We will consider both processes on the product space =
1
×
2
.
Thus we set F = F
1
×G
2
, F
t
= F
1
t
×G
2
, and P
x
= P
1
x
×P
2
. Moreover, we define
X
t
(ω) = X
1
t
(ω
1
), T
t
(ω) = T
2
t
(ω
2
), and θ
t
(ω) = θ
1
t
(ω
1
), where ω = (ω
1
,ω
2
) ∈
. Then X = (, F , F
t
,X
t
,θ
t
, P
x
) is a d-dimensional F
t
-Brownian motion, and
T = (, G,T
t
, P
x
) is an α/2-stable subordinator starting at zero, independent of
X for every P
x
. From now on, all processes and random variables will be defined
on .
Let A
t
= inf{s>0: T
s
≥ t} be the inverse of T . Since (T
t
) is strictly
increasing, (A
t
) is continuous. Further, A
T
t
= t and T
A
s
−
≤ s ≤ T
A
s
.
We define a process Y subordinate to X by Y
t
= X
T
t
. It is well known that Y is
a rotationally invariant α-stable process in R
d
.Ifµ
t
is the distribution of T
t
(i.e.,
(µ
t
,t ≥ 0) is the one-sided α/2-stable convolution semigroup), and (P
t
,t ≥ 0)
the semigroup corresponding to the Brownian motion X, then for any nonnegative
Borel function f on R
d
, E
x
(f (Y
t
)) = E
x
(f (X
T
t
)) = E
x
(
∞
0
f(X
s
)µ
t
(ds)) =
∞
0
P
s
f(x)µ
t
(ds).
Let D ⊂ R
d
be domain, and let τ
Y
D
= inf{t>0: Y
t
/∈ D} be the exit time of
Y from D. The process Y killed upon exiting D is defined by
Y
D
t
=
Y
t
,t<τ
Y
D
∂, t ≥ τ
Y
D
=
X
T
t
,t<τ
Y
D
∂, t ≥ τ
Y
D
where ∂ is an isolated point serving as a cemetery.
Let τ
D
= inf{t>0: X
t
/∈ D} be the exit time of X from D. The Brownian
motion killed upon exiting D is defined as
X
D
t
=
X
t
,t<τ
D
∂, t ≥ τ
D
We define now the subordinate killed Brownian motion as the process obtained
by subordinating X
D
via the α/2-stable subordinator T
t
. More precisely, let Z
t
=
(X
D
)
T
t
, t ≥ 0. Then
Z
t
=
X
T
t
,T
t
<τ
D
∂, T
t
≥ τ
D
=
X
T
t
,t<A
τ
D
∂, t ≥ A
τ
D
where the last equality follows from the fact {T
t
<τ
D
}={t<A
τ
D
}. Note that
A
τ
D
is the lifetime of the process Z. Moreover, it holds that A
τ
D
≤ τ
Y
D
. Indeed, if
s<A
τ
D
, then T
s
<τ
D
, implying that Y
s
= X
T
s
∈ D. Hence, s<τ
Y
D
. Therefore,
the lifetime of Z is less than or equal to the lifetime of Y
D
.
For any nonnegative Borel function f on D, let
Q
t
f(x) = E
x
[f(Y
D
t
)] = E
x
[f(Y
t
), t < τ
Y
D
] = E
x
[f(X
T
t
), t < τ
Y
D
]
R
t
f(x) = E
x
[f(Z
t
)] = E
x
[f(X
D
)
T
t
] = E
x
[f(X
T
t
), t < A
τ
D
]
Since A
τ
D
≤ τ
Y
D
, it follows that R
t
f(x) ≤ Q
t
f(x)for all t ≥ 0.
The following result was established in [18].
Subordinate killed Brownian motion 609
Proposition 2.1. Suppose that there exists C ∈ (0, 1) such that P
x
(X
t
∈ D) ≤ C
for every t ∈ (0, 1) and every x ∈ ∂D. Then
(1 − C)(1 − R
t
1(x)) ≤ 1 − Q
t
1(x) ≤ 1 − R
t
1(x) (2.1)
for every t ∈ (0, 1) and every x ∈ D.
A domain D ⊂ R
d
is said to satisfy an exterior cone condition if there exist a
cone K with vertex at the origin and a positive constant r
0
, such that for each point
x ∈ ∂D, there exist a translation and a rotation taking the cone K into a cone K
x
with the vertex at x such that
K
x
∩ B(x, r
0
) ⊂ D
c
∩ B(x, r
0
).
Here B(x, r
0
) denotes the ball of radius r
0
centered at x. It is easy to show that
the condition in Proposition 2.1 is true for a any domain D ⊂ R
d
satisfying an
exterior cone condition. It is well known that bounded C
1,1
domains and exterior
C
1,1
domains satisfy the exterior cone condition.
Let q(t, x,y) = q(t,y−x) be the transition density of the rotationally invariant
α-stable process Y . It is well known that the transition semigroup Q
t
corresponding
to the killed stable process also has a density. Let q
D
(t,x,y)be this density. Let
r(t,x,y) be the density of R
t
and let p
D
(t,x,y) be the transition density of the
killed Brownian motion X
D
. The density r(t,x,y) is given by the formula
r(t,x,y) =
∞
0
p
D
(s,x,y)µ(t,s)ds, (2.2)
where µ(t, s) is the density of the one-sided α/2-stable convolution semigroup
µ
t
. Let G
D
(x, y) and G
Y
D
(x, y) denote the Green functions of Z and Y
D
respec-
tively.The Green function of Z is given by
G
D
(x, y) =
∞
0
r(t,x,y)dt =
1
(α/2)
∞
0
p
D
(t,x,y)t
α/2−1
dt . (2.3)
The following elementary result was shown in [18].
Proposition 2.2. Let D be a domain in R
d
.
(i) The transition density r(t,x,y) of Z is jointly continuous in (x, y) for each
fixed t. Further, r(t,x,y) ≤ q
D
(t,x,y)for all t>0 and all (x, y) ∈ D ×D.
(ii) When d>α, the Green function G
D
(x, y) is finite and continuous on D ×
D \{(x, x), x ∈ D}.
It is well known (see, for instance, Example 1.4.1 of [10] and (2.20) of [2]) that
the Dirichlet form (E
Y
, F ) associated with Y is given by
E
Y
(u, v) =
1
2
A(d, −α)
R
d
R
d
(u(x) − u(y))(v(x) − v(y))
|x − y|
d+α
dxdy
F =
u ∈ L
2
(R
d
) :
R
d
R
d
(u(x) − u(y))
2
|x − y|
d+α
dxdy < ∞
,
610 R. Song
where
A(d, −α) =
α(
d+α
2
)
2
1−α
π
d/2
(1 −
α
2
)
.
If D is a domain in R
d
, then the Dirichlet space on L
2
(D, dx) of the killed
rotationally invariant α-stable process Y
D
is (E
Y
, F
D
) (cf. Theorem 4.4.3 of [10]),
where
F
D
={f ∈ F : f = 0 q.e. on D
c
}.
Here q.e. is the abbreviation for quasi-everywhere with respect to the Riesz capacity
corresponding to the process Y .Foru, v ∈ F
D
, E
Y
(u, v) can be rewritten as
E
Y
(u, v) =
D
D
(u(x) − u(y))(v(x) − v(y))J
Y
(x,y)dxdy
+
D
u(x)v(x)κ
Y
(x)dx,
where
J
Y
(x, y) =
1
2
A(d, −α)|x − y|
−(d+α)
, (2.4)
κ
Y
(x) = A(d, −α)
D
c
1
|x − y|
d+α
dy. (2.5)
J
Y
and κ
Y
are called the the jumping and killing functions of Y
D
respectively.
Z is a symmetric Markov process and so there is a Dirichlet form (E ,D(E))
associated with Z. Let P
D
t
be the transition semigroup corresponding to the Brown-
ian motion killed upon exiting D and recall that the corresponding transition density
is denoted by p
D
(t,x,y). It follows from [3] and [15] (see also [12]) that the jump-
ing function J(x,y) and the killing function κ(x) of the process Z are given by the
following formulae respectively:
J(x,y) =
∞
0
p
D
(t,x,y)ν(dt) (2.6)
κ(x) =
∞
0
(1 − P
D
t
1(x)) ν(dt), (2.7)
where
ν(dt) =
α/2
(1 − α/2)
t
−α/2−1
dt
is the L´evy measure of the α/2-stable subordinator.
It is easy to see from (2.6) that J(x,y) ≤ J
Y
(x, y) for every x, y ∈ D. The
following result, proven in [18], shows that the killing functions κ(x) with κ
Y
(x)
are comparable.
Proposition 2.3. Suppose that there exists C ∈ (0, 1) such that P
x
(X
t
∈ D) ≤ C
for every t ∈ (0, 1) and every x ∈ ∂D. Then
(1 − C)κ(x) ≤ κ
Y
(x) ≤ κ(x), for every x ∈ D. (2.8)